"One night we are driving somewhere around the city and hit a few potholes," he explains. "Just like that, we jokingly said, 'well, Jesus, maybe you should be the mayor, change something'. The joke went like this for a couple of weeks and then suddenly I said, 'why not?' - and it started getting serious."
It is still difficult to imagine Nastase being altogether serious about anything. This, after all, is the man who even in the peak of his tennis playing days was equally famous for his extraordinary antics and temper tantrums on court and wild lifestyle and womanising off it.
And although he will turn 50 later this year, much of the old rebel, with the long black hair and the mischievous glint in his eye, remains.
"Sure, it is not going to be easy getting up at seven in the morning and having to put on a tie to go to work," he concedes, "but if that's what I have to do I'll do it. Similarly with people, I used to get very upset with them but now I will probably have to control myself more. It will be a challenge, but I have known that before, when I was 18 and left Romania to go and challenge everybody in the world."
Bucharest potholes apart - and there are enough to occupy any future mayor of the city for decades - Nastase's decision to launch a career in politics stems from his growing disillusionment with life as an ageing ex-tennis star. He has been reduced to exhibition matches with old pals such as Jimmy Connors, and jetting between his various homes in Romania, France and the United States to give lucrative, but uninteresting coaching lessons to high-flying executives.
"I used to think I would want to play tennis for ever," says the former world number one, relaxing in the lounge of his luxurious Bucharest villa. "But about a month ago I suddenly felt I no longer wanted to get dressed in shorts and go running after a little ball - and not even catching it. I felt fed up. Maybe it was mid-life crisis."
Nastase's quest for a new role coincided neatly with the urgent need of Romania's ruling Social Democracy Party (PDSR) to resuscitate its flagging political fortunes after six years in charge of a country still struggling to emerge from the horrors of the Ceausescu era.
The party, the direct descendant of the old Communist Party, had already recruited a host of pop stars, actors and sports personalities to jazz up its image when, to its delight, it captured the support of Nastase - a man who still commands almost universal respect and admiration.
Late last year, Nastase decided to join the party in what was vaguely described as a "senior capacity". By January the PDSR, which faces national and presidential elections this year, rewarded him with the party nomination for the prestigious post of Bucharest mayor in elections due in May.
With an astonishing lack of dissimulation, Nastase admits that the main reason he joined the PDSR was because he took a liking to party president Adrian Nastase (no relation). Had someone else contacted him earlier, he may well have joined another party, he adds
Policy is not his strong point. Asked about what he would do if elected, the man who would be mayor speaks vaguely about taking one of the city's many disaster areas - roads, water and heating supplies, unfinished building projects - and "starting from zero, make it new. Then next year we have one less problem." He would like Bucharest to be clean - as would the two million people who live there - and may appeal to citizens to roll up their sleeves and voluntarily tidy up around their shops, streets and houses. In return he suggests, they might be given free entry to city cinemas.
Nastase also plans to ask for a little help from his foreign friends, among whom he counts French President Jacques Chirac and the mayor of Los Angeles, Richard Riordan.
If elected, Nastase, who despite his many years abroad never gave up his Romanian citizenship, plans to move his second wife, Alexandra, and two children to Bucharest, possibly for good.
On the grimy streets of the capital, people wading through the sludge and snow have mixed feelings about the mayoral aspirations of their most famous son.
"We will all vote for him because he started from the bottom and knows the hard life," says Ghilea Petrica, a shop assistant. "Maybe he will manage to turn this city back into the 'little Paris' it once was." But others were less favourably disposed towards him. "Nastase was a great tennis player and should have stuck to that," says a market-stall trader, Dan Bordea. "He should not have got mixed up in politics and especially not with the PDSR. He does not have any experience of politics and should leave it to the professionals."
But who are the professionals in post-Ceausescu Romania? Few would dispute that many of the country's officials have become corrupted in office or that the current mayor of Bucharest, Crin Halaicu, a member of the opposition Democratic Convention, has been a disaster.
Campaigning on a slogan of "Nastase - an Honest man", he wants to breathe some fresh air into his country's political life and offer some alternative to the normal diet of grey men, stifling bureaucracy and seeming stagnation.
It may not mean that all the potholes get fixed, but it could give the city back some of its former colour.
It might also bring some life back to politics. At an early campaign rally, an elderly voter pressed the would-be mayor to reveal how many millions he had made and stashed away.
"I don't know. I never stopped to count," replied Nastase, flashing a quick smile. "But I'll tell you one thing - I've certainly spent a few."