When Neil Kinnock bounds up after the football to say hello, I want to throw my arms around him and kiss his beaming face. He has just watched his Cardiff team lose to Eddie Izzard's beloved Crystal Palace, but anyone watching the two men would think it was the other way round, for the comedian still looks as blankly distant as he has all day.

At first I'd put Izzard's humourless demeanour down to pre-match nerves. A lifelong Palace fan, he became an associate director of the club last year, so we meet for lunch at the ground before the game. By the time post-match drinks arrive, five hours later, I have to conclude that while I had been very much looking forward to our meeting, he evidently hadn't. Then up bounces the former Labour leader. It's a relief just to see a friendly, smiling face – but all the more so when it turns out that even with Kinnock, one of his all-time heroes, Izzard remains detached. Nothing seems to defrost the comedian, which makes me feel better about the whole strangely tense day – but even more confused about Izzard.

Now 51, he has been widely adored for more than two decades. His fans worship his surreal, Pythonesque humour, and few were put off when he came out in the 90s as a heterosexual transvestite, and began cross-dressing on stage. Everyone else fell in love with him in 2009, after he ran 43 marathons in 52 days for Sport Relief. This year he performed the biggest-ever global standup tour, selling out from South Africa to Zagreb with a routine covering everything from human sacrifice to Olympic dressage, and the DVD – Force Majeure – has been enthusiastically reviewed. His childhood ambition to be an actor has come true, with roles in Ocean's Twelve, The Avengers and other films. Most impressively of all, the French leg of his Force Majeure tour was performed entirely in French. Izzard is now studying German, and once he's pulled off a performance in that, too, he'll move on to Spanish, then Russian.

So why make such heavy weather of an afternoon at the football? It starts with an innocent question about if it's harder to be funny in a foreign language. "No, the difficult thing to learn is how to have a sense of humour. Language is not difficult to learn." But even when I used to be pretty much bilingual, I say, I still wasn't very funny in French. He studies me doubtfully. "Well were you – are you – funny in English?" Funnier than in French, definitely.

"Well, you didn't try it. I mean, I made a guy laugh in a traffic jam at the Cannes film festival. We were going along at nought miles per hour. Then after about 20 minutes we went up to 5mph. So I said, 'Wow – c'est comme le Grand Prix maintenant.'" I ask what he'd have said to the driver had they been in England. "Er, probably, 'It's like Brands Hatch now.'" I think that's a bit funnier. "No, not in France." But that's my point – surely humour often lies in the specificity of a particular reference? "Look, it doesn't really matter," he says curtly. "I do universal humour." Eventually, he says, he will learn Arabic, too. "I was born in Yemen, so it's my duty to do this when Arab people are struggling so hard to try to get democracy going. I have to play Yemen. This is right that I should be doing this."

By May 2019, however, Izzard intends to be winding up his showbusiness CV and launching a political career. "Everybody says I would be a good fit for mayor," he explains, "because it's a personality-led thing," but if by then London has an incumbent Labour mayor, he'll run for parliament instead. Having helped Labour fight four general elections, and been cheerleader-in-chief for Ken Livingstone's 2012 mayoral campaign, he has a good idea of what he'd be getting himself into. "But I have to do this in my life. I should stand up and be counted."

Then home secretary Alan Johnson (left) and Eddie Izzard on the general election campaign trail in 2010. Photograph: Owen Humphreys/PA

It's a laudable impulse, but while I was watching him wow Wembley on his new DVD, I found it quite hard to picture him planning local bus routes with a transport committee. I can't think of any precedent. "There is in America," he snaps. "Al Franken's done it. Who I know well. I said to him, 'What you're doing is what I'm trying to do.' And he's already done it."

Franken's comedy was always political, though, and Izzard's emphatically isn't, so I try to get a sense of his beliefs. He joined the Labour party in 1995, and his politics sound like vintage New Labour: "Safety net for people who are having a tough time, encourage wealth creation on the positive side." He hates extremism, and as a "radical centrist" thinks he has a unique political advantage.

"You see, I've never heard anybody else talk about this, but I think I do know where the centre of an argument is. And I think that's an important thing. I've heard people arguing, and you think: actually, do you two realise you're fighting like crazy but you're actually saying the same thing? So I want to get involved. I do do things in a different way."

I'm not sure that all political arguments can be squared so easily. Take the London riots, for example. According to the right, the rioters were greedy yobs; to the left, they were victims of a consumerist society. Where would Izzard locate the central consensus in that argument? "It's probably somewhere in between them, isn't it?" he retorts. "No, I haven't analysed that. I wasn't mayor of London at the time. I haven't analysed exactly what was happening, no. When I get in, that's when I'll make my decisions."

He's oddly evasive about where he lives. He does own a flat in London, but rents it out, and says he stays in a hotel when he's in town. But where is all his stuff? He hesitates. "Er. My stuff is in … places." Izzard has always refused to discuss private relationships, so may just be trying to avoid mentioning a girlfriend. But he goes on, "I don't really need stuff. Some people want to amass stuff, but that's not really what I'm interested in."

He looks surprised when I ask why he wants to be mayor. "Why? I think I can put a lot of energy into it. I was born in London, I think London's a great city. I think I can bring unusual things to the table, and think in a different way. And, ummm … I mean, why shouldn't I be?" One reason could be that as a rich celebrity who spends a good deal of his time abroad, he doesn't face many of the daily issues most Londoners do. He looks annoyed, and too late I realise his question had been rhetorical.

"But I have been through lots of people's experiences in London, when I was living and working in London. I have now created this comedy, which is my business, and exported it around the world. I cannot get it around the world without going there. I'm not making widgets, so actually, unfortunately, I have to take it around. I think it's a positive thing if someone like myself, who's openly, you know, out as a straight transvestite, who's played the Hollywood Bowl, played Madison Square Garden, who is touring around arenas doing gigs in French, German – I think that's pretty positive. So any kid – girl, boy, straight, gay, lesbian, doesn't matter – they could say, 'Well, that person did OK, by working their way up from the streets of London.'"

It's hard to work out what he would actually want to do in office. If he could take his pick of Whitehall, which government department would he love to run? After 10 seconds of silence, "I am not sure. That's a bit like asking me what my dream acting project would be. I don't think I quite work in that way." I'm just trying, I explain, to get a sense of his key priorities. "Well, politically, I want myself to do well, and I want other people to do well.'"

Eddie Izzard as he completed the 40th of 43 marathons he ran in 52 days in aid of Sport Relief in 2009. Photograph: Geoff Pugh/Rex Features

Izzard describes himself as a "glass two-thirds full" optimist – "You know, I'm a transvestite who's got this far, so you've got to be pretty positive" – and his political approach as "an open hand, where you say, 'Who are you? Hi. Can I shake your hand?'" But although he is polite to everyone we meet all afternoon, I don't spot any conspicuous embrace of humanity. It's easier to see why he was drawn to the loneliness of long-distance running, and no great surprise to read that he once said, "I don't really have friends." If he wants to be a politician, he'll need to learn how to look as though he's listening when people talk to him, instead of staring vacantly into space. I worry, too, about how he will cope with the hatred heaped upon every politician.

"Well, how much hate," he flashes back, "did I get for being a transvestite? It's very, very high. I've had things said to me that … " and he breaks off, as if the insults are simply unrepeatable. Such as? He steels himself. "Like: 'What the fuck is that?'" – and fixes me with burning stare. "You know, if I can deal with that, then I think politics is easy." It's not an entirely reassuring response.

Izzard has always described himself as a transgender heterosexual who favours a "girl look" when he's feeling in "girl mode", and a "boy look" when in "boy mode", saying of his genetic makeup, "I just have an extra girlie bit, which surfaces with the heels, skirts and makeup." So I ask if he will campaign wearing women's clothes.

"I don't wear women's clothes," he corrects sharply. "Are women transvestites because they're wearing trousers? No. They just say they're wearing trousers. So I deny I'm wearing women's clothes. I just say I'm wearing clothes." How would he like me to refer to them? "Just say, 'wearing dresses', or 'wearing makeup'."

Would he campaign wearing a dress and makeup? "I'll campaign with these nails," and he holds up a long, immaculately painted set, blood-red save for a union flag on one finger, and an EU flag on another. "If I've got my nails painted then that's as much girl mode as anything." It may be to Izzard, but for many voters I'd guess that a male mayor in a dress is a different proposition to one with red nails. "No, not really," he counters coldly. "I'm saying that I will wear whatever I want to wear at whatever time. I don't know what you're trying to gauge there." He was the one who kept citing transvestism in the context of his future career, so I don't know why he minds me trying to establish what voters can expect. For several years he has referred to going through a "blokey phase", and I can't recall the last time I saw a picture of him in anything more adventurous than nail polish, mascara and heeled boots, so I ask when he last put on a dress. "I'm not sure," he says, in a tone that makes it sound like, "I'm not saying."

It takes a lot to walk away from global stardom and artistic freedom. To give it all up for the dreary discipline of politics is astonishing. I wonder how he'll cope with the relentless clamour for his undivided attention, or satisfy voters' need for a sense of personal connection. Izzard isn't worried at all.

"I have always learned how to do things while struggling at the beginning to work out how they work. I started in sketch comedy, then I was a street performer, then I was a standup, then an actor, and I wasn't good at any of them at first. So I'll probably screw up. But I'll learn."

• Note: please see our reply to comments regarding Eddie Izzard's place of birth