Mr Johnson is soon to publish his 2020 vision for London, a collection of policies designed to make the city more attractive to business and global investment. “The vision we have for London is a city that is not just a world leader in finance or arts and culture. I want London to be a scientific and technical capital, too, with more spin-off businesses from universities. We must talk about enterprise – not austerity.”

Big spending on infrastructure is key to this plan, with a second Crossrail across the capital, longer tube lines and more river crossings. There must be a swift solution to the capacity crunch above Britain’s skies and a new hub airport. He considers expanding Stansted just as viable as building a new airport in the Thames estuary, aided by a sovereign wealth fund from China or another developing world economy.

“You just have to chuck a snowball into a cocktail party at Davos and you’d hit someone with a sovereign wealth fund who would fund a piece of infrastructure like that,” he says.

He is less ebullient about the Government’s proposed HS2 rail link between London and the North. He says he is “very worried” about how the existing tube lines will cope with the extra human traffic and impact on the suburbs.

“This vision is about making us open to the global economy, with low taxes, a 24-hour hub airport and sensible visa rules.”

But hang on, doesn’t his pitch fly in the face of the plans unveiled last week by David Cameron to offer an in-out referendum? Some well-known business leaders – including Sir Richard Branson and Sir Martin Sorrell – have expressed fears that Britain’s future in the single market will “create damaging uncertainty for British business”. Labour have echoed these concerns, warning that plans for a possible exit from the EU will be a drain on inward investment to the UK.

Mr Johnson teases businessmen with such worries. “Obviously, if businesses had their own way, they would get rid of all democratic consultations,” he says. “That way there would be no danger of Labour getting in and whacking up taxes, but that doesn’t mean we’re going to abolish general elections.

“Britain is the most flexible place to invest. Is the tax rate right? Is the workforce competitive and well-skilled? Is the logistical stuff good? Is the crime rate low? We can say yes to all those questions. That’s the real certainty business want.”

He argues that the EU has changed “out of sight” since the last referendum in 1975 and there is clear public appetite for such a vote now. “It’s high time the people were consulted,” he says.

Mr Johnson has stern words for the French foreign minister Laurent Fabius, who last week said that Britain is behaving like someone who joins a football club and promptly demands to play rugby. It’s a sentiment shared by many European politicians who accuse Britain of trying to morph the EU into “à la carte” or “pick and mix” union.

“What utter nonsense! We joined a football club where a minority of the members are now trying to play a peculiar game of their own which is going very wrong. And they are making up the rules as they go along.”

He is referring to the closer fiscal integration of the eurozone economies. His position is at odds with that of Mr Cameron and George Osborne, who feel that such a “big bazooka” solution is the only way to solve the eurozone’s debt crisis.

Mr Johnson clearly signals that he will vote to leave the EU if Britain’s relationship with the single market remains unchanged. “I will vote yes if we get the right deal, but obviously logically I can’t rule out voting no.”

So how many powers need to be repatriated for him to vote for Britain to stay in the single market? He sets out three targets: employment regulations, fisheries policy and the EU social chapter.

In fact, he thinks Britain is unlikely to leave the single market.

He argues that Germany, Holland and some Scandinavian countries share UK concerns about how the EU has changed and will be willing to grant concessions. “We’ve got plenty of friends in Europe who still want us as a powerful voice to speak up for reform,” he says, adding that allowing Britain some leeway will also “be a relief” for other European countries.

“If they give Britain a deal we are happy with, they won’t have to endure the misery of us constantly complaining about things in the future.”

To underline his point, he cites Neelie Cruz, an EU commissioner who raised the prospect of introducing Europe-wide regulation of newspapers while at the Davos summit.

“Donnez-Moi un break – as we used to say in Brussels,” roars Mr Johnson. “Who does she think she’s kidding? We’ve got this poor man Leveson wrapping a towel round his head trying to work out how on earth you should regulate the press at a national level. Now there’s someone in Brussels trying to do the same.”

Yet it appears too late to renegotiate one edict from Brussels that is alarming many in this country. From December, Britain will have to open its borders to any Romanians and Bulgarians who want to settle here.

Mr Johnson says this “unquestionably poses problems” for London, which already has what he describes as “massive pressures” on housing and school places.

“In an ideal world, we would have a points system or way of making sure those who are coming here are economically active and not a drain on resources,” says Mr Johnson. “We need to think about this – especially considering the underestimate in the number of people who came from Poland and the other accession countries the last time.”

Official estimates now suggest that somewhere between one and two million eastern Europeans came to Britain after migration restrictions from Poland and seven other European countries were relaxed in 2004. The Home Office had initially expected 13,000.

However, there seems little the Coalition can do to stop a large influx this time.

“It is worth a conversation and it is worth thinking about,” he says, arguing that taking some steps to check this type of immigration might assuage some Eurosceptics.

He argues that uncontrolled immigration is “one of the things that has most unnerved” the British about Europe, and that doing something about this might make the British public more likely to vote to remain in the EU.

Amid the weighty discussions of Davos, Mr Johnson did find time to enjoy a dinner on Thursday evening with the Prime Minister and Mr Osborne. Onlookers said the trio were in high spirits – a fact that offended some commentators, as the trio might have been aware that official data awaiting release the next morning would confirm that the economy shrank during the final three months of last year.

“Well, I would say I didn’t stay long… I had to go and meet the Malaysian prime minister,” says Mr Johnson, who didn’t, in fact, know the GDP numbers.

“But it was not opulent – and not paid for by the taxpayer. It was a very jolly evening. And fairly raucous, yes.”

Thank goodness the new Serious Boris retains a sense of fun.