Bill Bailey is exasperated. He is trying to make Geraldine McNulty share his memory of the moment he held her for the first time. But will she? She will not. He tells her he held her breasts on a bridge as they paused during a walk. It is the most important memory of his life, but she looks blank, which - understandably, perhaps - enrages him even more. For her, there was no bridge and their first tryst was somewhere else. She recalls not how he touched her breasts, but how he took her hand and gently stroked it.

McNulty and Bailey are rehearsing Harold Pinter's 1969 sketch Night in a room near the glamour of the Blackwall Tunnel Approach Road in east London. It is one of 14 sketches that the Nobel laureate has written over the past half-century that Bailey, the stand-up comedian and TV performer who once called one of his shows Part Troll to satirise his endearingly woodlandy persona, is to stage in the West End next month. They deal with Pinter's enduring dramatic themes - derangement, erotic fantasy, obsession, family history, nameless menace.

There is a touch of Lerner and Loewe's song Ah Yes! I Remember It Well about the sketch McNulty and Bailey are rehearsing. But only a touch: this is hardly sentimental comedy but a dramatic gem about how a husband and wife differ in the memories of their first moment of intimacy, and how a horrible tragic chasm of fundamental misunderstanding suddenly opens up only to be quickly closed over - the kind of closure perhaps necessary for such a relationship to endure so long.

It is only five pages and yet Night maps the emotional terrain of a long relationship convincingly. "Has anyone ever caught more finely the differences between male and female feelings about heterosexual love than Pinter?" critic Alastair Macaulay asked when reviewing Night. Quite possibly not.

But it is also very funny, and this is why Bailey is drawn to Pinter's writing. When McNulty looks wonderfully disbelieving and says, as if puzzling over dubious calculations on a tax return, "Were they in your hands? My breasts?", I hoot embarrassingly.

Bailey even shakes his cupped hands in front of McNulty, as if willing her to remember that he did in fact fondle her on the bridge. It is at this point that the director, Sean Foley, yells out sensibly: "Don't milk the breasts!" His point, of course, is that Pinter does funny, but if you push too hard in that direction all the other things he dramatises - tragedy, tenderness, poignancy, passion and pain - get lost.

No matter. Night slayed them in Ilfracombe, Devon, where Bailey, along with comic actors Kevin Eldon, Sally Phillips and McNulty, tried out their production recently. Next month, the aim is to slay them at the Theatre Royal Haymarket in London.

Many of the sketches were written for theatre or TV revues. "You'd have a song and a dance and then a sketch by Pinter," says Bailey. "What a night!" After the revue died as a form of entertainment, Pinter carried on writing little gems - often about political themes such as torture or totalitarianism - that have occasionally been staged. But this will be the first time that all these sketches, brought together by Bailey under the title Pinter's People: Sketches, Monologues and Two Handers 1958-2007, will be performed together in an evening.

Is this move into Pinter part of the diversification of the extraordinarily successful international Bill Bailey brand? "Well, I've become successful and that's created opportunities, I suppose. It's also changed me. I'm actually worried because I've started to refer to myself in the third person. When I get asked questions in interviews I ask myself internally, 'What would Bill Bailey say?' It's a bad sign. Maybe Bill Bailey's lost it.

"Realistically, the fact that I've got a higher profile now has made this production of Pinter happen more easily. It's the way of all things and I'm aware of that. So I'm aware that it has to be good - I don't want people to think I'm just doing it for the sake of it. It's something that's been dear to my heart for years and now I'm in a position where I can pull it off. I'm very grateful for that."

Bailey should not really be a bearded global comedy brand, covering the many platforms of sitcom (Black Books), nature show (Wild Thing - I Love You), stand-up (his American tour is looming, his Edinburgh gigs an annual ritual), music (he is a multi-instrumentalist with perfect pitch who once memorably performed a Kraftwerk homage entitled Das Hokey Cokey), Pinterama and an actorly prehistory in which he claims to have interpreted the role of a disenfranchised owl for a Welsh theatre troupe.

No, Bailey should really still be working in telesales. He would have been, too, were it not for his boss ("let's just call him the Man," says Bailey) insisting that he wear a tie to work. "I could never understand that: who needs a tie for a job that involves Working. On. The. Phone?" He resigned on this sartorial principle and went back to trying to make his poorly attended stand-up shows work.

It proved a good career move for the Bath-born 42-year-old (real name Mark; the Bill only came when a history teacher sang the trad jazz standard Won't You Come Home, Bill Bailey? in class) who rose to stardom in the mid-90s. In 1995, his show Bill Bailey's Cosmic Jam so successfully showcased his befuddled hippie persona and musical virtuosity that he was commissioned to make a BBC TV special called Is It Bill Bailey? In 1996, he narrowly lost the Perrier award at Edinburgh to Dylan Moran. Since then he has become very nearly ubiquitous: a guest on QI, a turn in the film of The Hitchhiker's Guide to the Galaxy, appearances on Room 101 and Top Gear, a cameo in the looming Britflick Hot Fuzz, not to mention straddling the Atlantic and reaching Down Under with his stand-up comedy.

We are in the cafe of the Three Mills Studios, where Robbie Williams has recorded, some of cinema's great directors (Loach, Leigh, Cronenberg) have filmed and thesps daily finesse their lines. It is near the site of the 2012 Olympics. Outside, the sun has set bewitchingly over Tesco and the River Lea. Inside, part of me longs to reach over with a pair of scissors and trim Bailey's trademark mullet, which would be rude and ungracious: Bailey is engaging company.

He tells me he has wanted to stage Pinter's sketches together for 20 years. "I first read them when I was 19 and they've stayed with me." He was then a London University undergraduate studying English and drama, rehearsing a production of Pinter's play Night Out. "At the back of the anthology we were working from were these sketches and they captivated me. They rang the same bell in my head as Alan Bennett would do in his detailed monologues and the way he gets people to trip themselves up on their own speech.

"So much of what was supposed to be serious drama left me slightly cold. I always thought I was appreciating drama in the wrong way. Drama's not supposed to be funny and it's not supposed to mimic ordinary life or the naturalistic way people talk. It's supposed to have an elevated sense of prose. But these immediately struck me as being something more accessible."

Bailey says he experienced an echo of these feelings when he spoke to students after they saw the Ilfracombe performance of Pinter's People. "They were saying that one of the sketches was a set text but that when they read it in the classroom nobody was laughing. When they came to see it they were falling about - they had never thought to see it the way we did it. That's 'Job done' for me."

Some of the sketches date from the late 50s. "Pinter wrote one for the BBC called Trouble at the Works about the power relationship between a boss and a trade union official. I could see how that would be funny on the stage immediately."

But, again, this sketch was not just funny - it was also political dynamite. "The BBC made Pinter change the punchline because they thought it would foment social unrest, which seems quite incredible now." The union official tells the boss that his men no longer want to make an abstruse machine tool. "What do they want to make in its place?" asks the boss. "Trouble," says the union man in reply. "It's a great line, but they made him change it to 'brandy balls'. When we met him he was still angry about it, even though it was 47 years ago. He would just say darkly: 'The BBC made me change it ... ' I felt like saying: 'Let it go Harold,' but of course I didn't."

How did you get on with Pinter? "With a mixture of terror and awe. We performed the sketches for him at the Arts Theatre, and he sat in the circle. He said, 'This is the exact seat I sat in for the opening night of The Caretaker.' So, no pressure! I was terrified. But after the performance he said, 'I found that ... (Yes, what?) ... absolutely ... (Oh God, he hates it!) ... riveting.'" Thus did the Nobel laureate give the stand-up his imprimatur. "Luckily, he was very positive and effusive. Thank God!" Better yet, Pinter agreed to supply a new sketch for the evening. It turns out to be a two-hander about his mobile-phone aversion called Apart From That which was performed last June on Newsnight by Pinter and Rupert Graves, but still.

Bailey will even step into Pinter's shoes when he performs the role of a minister of culture in Press Conference. Pinter played that role at the National Theatre in 2002. The minister for culture is standing outside some premiere in his white tie and tails with a glamorous wife and he is telling the press that, yes, he was previously the head of the secret police. A reporter asks: "Do you not think there's a contradiction?" The minister replies: "None whatsoever. As head of the secret police, I was responsible for safeguarding our cultural inheritance from the forces intent on subverting it." Then he is asked about the state attitude to children and women and he replies, "We distrusted children if they were the children of subversives. We abducted them and brought them up properly. Or we killed them." "It's one of the funniest, most harrowing pieces of writing I know," says Bailey.

In another sketch called the New World Order, two men circle a hooded and bound man. "It's just so sinister and terrifying without any physical torture being inflicted," says Bailey, who will play one of the men. "Imagine picking that for the Royal Variety Show!" he says. "'And now a sketch about torture by the popular playwright, Harold Pinter ... ' The look on the Queen's face!" He mimics HRH: "Get Joe Pasquale back! This Pinter - he's a little too real!"

With all due respect, where do you get the nerve from to perform Pinter? "I suppose there's a feeling that comedians shouldn't be in the West End, shouldn't be actors - how dare they! What I'd say is that comics are used to doing this kind of thing. You've got to go on and be funny and get the timing absolutely right. That's what I do as a stand-up and on TV."

In 2002, he replaced Irish comic Sean Hughes as a team captain in BBC2's pop quiz Never Mind the Buzzcocks. The other week, the show offered him the chance to do a po-mo critique of celebrity fatuousness. It happened like this. Preston, the troubled crooner from ska revivalists the Ordinary Boys, was on Bailey's team. "He'd been on before and seemed quite nice but in the intervening period he'd gone on Celebrity Big Brother and met his wife Chantelle." You mean the ordinary girl who wasn't a celebrity but now is and was ordinary but now isn't? Bailey looks at me and rolls his eyes by way of affirmation. "Anyway, Simon [Amstell, the host] started teasing him by reading out chunks from her autobiography, Living the Dream, which, God bless her, isn't Thomas Pynchon. In terms of slaggings-off it was a one on the Richter scale - really gentle. And then Preston said of the autobiography, 'I haven't read it.' Which was an ungallant thing to say.

"He got the hump because we had the audacity or temerity to have a go at him. He ripped the mic off, said, 'Out of order' and stood up to leave. I managed to say that he thought he'd been voted off. That got a big laugh and he growled even more and left. He'd just been on a reality show, for God's sake! Any friend could have told him, 'Expect to get it with both barrels.' But he didn't."

Then Bailey had a brainwave. He walked into the audience and grabbed someone who looked like Preston. "He was smartly turned out with a quiff. So I said, 'You'll do.' And he was great. He was a really good sport. His family were there, laughing and loving it." The one-time ordinary boy replaced by the genuine article - nice touch.

It sounds just the sort of raw material that Pinter might have fun with over five blissfully bileful pages. But maybe Pinter has already written something like that. Earlier, I saw McNulty rehearse Request Stop, a sketch from the 60s featuring a drunk and deranged woman growling at a bus queue which wisely keeps its silent counsel. It is a situation that will resonate with anyone who has ever used public transport. She loses her temper with everyone for not sharing her delusion that she has been grotesquely insulted by a man in the queue, a delusion surely akin to Preston's misplaced offence on Never Mind the Buzzcocks.

"Excuse me, lady," says McNulty, tipsily. "I'm thinking of taking this man up to the magistrate's court. You heard him make that crack. Would you like to be a witness?" The woman steps into the road and sensibly hails a taxi. A few seconds later, the others in the queue, whom she bizarrely attacks for being Peruvians ("Don't worry, I know all about people like you"), run off stage after a passing bus, desperate to escape the woman and her potty worldview. True, Request Stop is not about celebrity, but it echoes that culture's narcissistic delusions. It is also, like all the Pinter sketches Bailey has assembled, disturbingly hilarious.

· Pinter's People will be at the Theatre Royal Haymarket, London, for a four-week run from February 1. Box office: 0870 380 2003.