The BBC likes to talk about journalistic excellence and a tradition of creativity, but really it is an academy for stars. And after Eddie Mair's careful pinioning and dissection of Boris Johnson on Sunday's The Andrew Marr Show, there is a feeling out there that a new one has just graduated.

In truth, the questions Mair raised over Johnson's integrity were far from new, and neither were they especially serious, but it was the first time the London mayor's bluffness has so publicly been called. As always, Mair was calm, empathetic even, but also painfully direct, saying things such as: "Let me ask you about a barefaced lie" and "You're a nasty piece of work, aren't you?" "I thought it was rather brilliant," says Robert Peston, Mair's frequent collaborator on the BBC. "I think his genius is to make people feel comfortable, and then lob in the incendiary."

For some time there have been murmurs that Mair is being readied to replace Jeremy Paxman as lead anchor on Newsnight whenever it may be that BBC2's great hardman decides to stop brutalising politicians and heckling his bosses in order to spend more time with his fishing tackle. Now those murmurs are getting close to a campaign. It certainly would be a new direction for the programme, which needs one.

Last November, by chance, Mair presided over the episode in which Newsnight had to report upon itself, and he did a sensational job. Despite being just a stand-in, he filled his delivery with beautifully ironic glances and exasperated comments, which exactly captured the absurdity of his position, having to expose the cowardice and incompetence of the very people he was working with. For good measure, it also confirmed that he can wield wit visually. "Newsnight will be back on Monday. Probably," he finished off, without even a flicker of a smile. "The thing that he does, with nuance, is that you always know what's at the back of his mind, without breaching BBC impartiality," says Peston. "It's very clever."

Behind the scenes at Newsnight, the reaction to Mair's performance is said to have been mixed. Some staff felt he delivered more and harder kicks than necessary, and appeared to be taking pleasure in a moment that had not been put there for his enjoying. Whether this is so or not, it presents a pattern typical to Mair's career. What he has is a special talent for connecting with an audience, even (or indeed especially) when this comes at the expense of connecting with his colleagues.

Eddie Mair: 'Easily the funniest man on BBC radio.' Photograph: Rolf Marriott/BBC

On PM, his regular weekday gig on Radio 4, he is well liked, but felt to be a private and complex person, who often takes the programme in more obscure or eccentric directions than some would like. Yet in return, he gets something close to devotion from his audience, who give excellent feedback on his performances. Indeed, to them, this triumph over Boris Johnson is less the moment of his graduation than a costumed diploma ceremony. On radio, Mair's star quality has been recognised nationally since he first began hosting Midday With Mair on the newly launched 5 Live in 1994.

Born and brought up in Dundee, state-educated, the son of a lorry driver and a nurse, he was as obvious a broadcasting prodigy as you could ever find. There is a story about him, barely pubescent, broadcasting to his schoolmates over the playground PA system, and later appearing on his local commercial station, Radio Tay, among a group of 14-year-olds, entirely calm about his debut. One way or another, at 17 he had a job there, and was hosting shows on the channel when he was still a teenager.

Having twice not quite started university, he joined BBC Scotland as a subeditor at the age of 22, and soon began to host its flagship radio and television shows: Good Morning Scotland and Reporting Scotland. In 1993, when he was 28, he won a Sony Gold award for a new radio breakfast show, Eddie Mair Live. From that came his involvement in the launch of Radio 5 Live, and then of Broadcasting House on Radio 4. This early practice at life on air became the basis, perhaps, for his exceptionally adventurous and yet relaxed approach to broadcasting. Indeed, the most important thing to say about him is that he is very funny – easily the funniest man on BBC radio at the moment, including its comedians. He has become famous for a wacky yet deadpan wit, as well as for his love of long-running jokes, such as the loyalty card he announced for listeners to Midday With Mair, or for the feud he insists that he is having with Robert Peston. ("It's ongoing. We loathe each other," is the latest from his nemesis on that.)

One favourite Mair gag says everything about how good, and how unusual, he is. On Broadcasting House, he followed a weather report predicting oppressively hot weather with the words: "So don't forget to wrap up warm." Soon, this became a kind of subversive mantra, repeated whenever the mood took him after a baking forecast, or even just a mild one. He still does it, even in tweets. What's funny is the way that Mair imitates the cosy patter of presenters everywhere – including those on the desks next to him – who mix little phrases of faux-empathy into their chat. He is telling his audience that he is being real with them, and implying that the others are all versions of Alan Partridge.

Eddie Mair grills Boris Johnson on The Andrew Marr Show. Photograph: Jeff Overs/BBC/PA

Interestingly, he writes and broadcasts a great deal, yet he seldom agrees to interviews – and chose not to break the policy for this article. All the BBC would say on the subject of the Johnson interview was: "We believe this was a fair interview which took in issues facing London and the wider political landscape as well as looking towards tonight's TV portrait programme. As the documentary is biographical, exploring controversial episodes in the mayor's life was considered appropriate. Eddie's line of questioning attempted to elicit responses to direct questions that were not being answered."

When Mair does talk, however, he is very funny – some might say evasively so. Asked what he thought of Fi Glover, his successor at Broadcasting House, he simply said: "I don't listen to her. She's awful." On being told that Glover (whom he now calls "Fig Lover") finds him funny, he replied: "Fi is often heavily medicated. I'm not saying she doesn't make any sense at times, but you've got to make allowances." Being one of the first gay news presenters, it is possible that Mair senses the questions people plan to ask, and prefers – as Johnson does – not to discuss his personal life. Certainly he has never been noticeably outspoken on gay issues.

Quick as he was to succeed in radio, though, Mair has been slow to transfer that success to television. Now 47, he has still never had his own high-profile show. This may simply be because he isn't interested. Although he was interested enough to take on Reporting Scotland, then to host Time Commanders in 2003, and now for Newsnight and The Andrew Marr Show, while Marr himself recovers from a stroke.

It is also tempting to detect a knowing wink in Mair's jokes about how he is excluded from Newsnight's inner circle. In his Radio Times column last year, he pretended to share some of the secrets behind the show, including its green room, which has "a four-digit pin code for the door, which I haven't been given and 'never will'."

He has lost a great deal of weight recently as well, which – who knows? – might be part of a plan to spend more time on camera. "I think he's brilliant on telly," Peston says, "and I think if the BBC decided to give him more telly, that would be a rather brilliant thing." Would he like that? "I don't know. I've not really asked him."

If Mair does desire more TV work, and has been hitherto denied it, it is easy to imagine reasons why. He may be too sophisticated and too irreverent. On air, he loves to undermine the medium itself, and the people that he works with. Just a few weeks ago, in the Radio Times, he called The Andrew Marr Show "Sunday morning's most-watched-programme-that's-not-pretending-to-be-even-a-little-bit-about-religion". In TV, the sheer scale of the production tends to call for rather more decorum in a host.

In another sense, though, Mair offers great hope. Listening to his broadcasts, with their humility and knowingness, he sounds like the first news presenter who has – finally – absorbed the impact of The Day Today, which punctured the absurdity of news media nearly 20 years ago, seemingly without affecting them. And here the similarities with Boris Johnson himself are striking too. A man of wit and talent come to brush aside the earnest phoneys? A man who shields his deep ambition beneath a comical exterior? Perhaps on Sunday they were both exposed.