Sadiq Khan has long been known as one of Labour’s most pugnacious politicians: someone who likes to fight, and likes to win. The son of a bus driver, he became a human rights lawyer, entered parliament in 2005 and that same year was named newcomer of the year at The Spectator’s parliamentary awards.

He ran Ed Miliband’s leadership campaign in 2010 and led Labour’s fierce — and surprisingly effective —campaign in London last year. Now, his sights are set on reclaiming City Hall for Labour and persuading even those on the right that he is the natural heir to Boris Johnson.

‘I want Spectator readers to give me a second look,’ he says, when we meet in the House of Commons. He is not, he’s keen to stress, a lieutenant in Jeremy Corbyn’s army. He’s keen to ladle praise on Boris Johnson — a ‘great salesman for our city’ who made him feel ‘proud to be a Londoner’ during the Olympics. He even likes rich people. ‘I welcome the fact that we have got 140-plus billionaires in London; that’s a good thing. I welcome the fact that there are more than 400,000 millionaires; that’s a good thing.’ If you shut your eyes, it could be Peter Mandelson speaking. It is not what you would expect from someone who has always been on the soft left of Labour.

If elected mayor, he says, he would not attempt to taunt David Cameron’s government as Ken Livingstone once taunted Margaret Thatcher’s. ‘I’m not going to be somebody who puts a big banner up outside City Hall criticising the Prime Minister, he says. ‘As a Labour councillor for 12 years in Tory Wandsworth I saw the benefits of having to work with the Tories to get a good deal for my constituents.’

But this is all part of Khan’s ambitious strategy: he doesn’t just want to win, he wants to win big. He is confident about his own ability to run a campaign; to him the issue isn’t whether he’ll win — but how.

‘If we wanted to, we could just target those Labour voters and increase the turnout. We could win London just by doing that.’ But, he says, ‘That’s not the sort of mayor I want to be… I want to be everyone’s mayor.’ In particular, he wants to be that vanishingly rare thing: a Labour friend of business. ‘Bearing in mind who our leader is,’ he says, ‘it’s important to reassure the right people that he doesn’t represent all Labour thinking.’ Khan is clearly aware that his biggest vulnerability is being branded Corbyn’s candidate. He is eager to say he is not in regular contact with his party leader; the last time he saw him was when they had their photos taken together to promote the Living Wage more than a month ago.

But if Khan is so distant from Corbyn, why did he nominate him for the Labour leadership? To widen debate, he says. And he insists that he’s not to blame for Corbyn winning. Instead, he points the finger at the other candidates. To emphasise the point, he adds, ‘I’m sure if Andy, Yvette and Liz were here they would say they had a shocking summer.’

Khan would be the capital’s first Muslim mayor. And London, he says, is the one of best cities in the world in which to be a Muslim. He is adamant that he wouldn’t want his daughters to grow up anywhere else. ‘There are laws here to protect them being discriminated against. Laws here that allow me, if I want to, to do my ablutions and pray. I could wear black and grow a beard; if my wife chose to she could wear a hijab. I feel so strongly about this.’ He argues that ‘we should explain to people in Muslim majority countries that I am the West so when they think they hate the West they are hating me.’ But he thinks that government needs to do more to promote integration; that London can’t simply be allowed to develop naturally any more. ‘We’ve got to think a bit more about how we encourage, cajole people into integrating much more,’ he says.

Housing is, Khan believes, the ‘biggest crisis facing Londoners’. He fears that the capital is ‘being hollowed out’ and wants all new properties to have to be offered to Londoners before they can be sold to foreigners. He also proposes a London ‘living rent’ — essentially a form of rent control. This is a reminder that Khan, however moderate he now sounds, remains very much on the left.

While he is the son of a bus driver, Zac Goldsmith is the son of a billionaire. But Khan pledges not to make this point too often. ‘Criticising him for who he is and where he’s come from; I won’t do that,’ he says. Voters, he suspects, don’t much care if Goldsmith is an Old Etonian. ‘When you look at who the Prime Minister is, it didn’t do him any damage.’

The Tories would dearly love to turn this contest into independent-minded Zac versus Jeremy Corbyn’s man. But by love-bombing Tories and business, Khan is determined to stop them doing that. So if the Tories are to stop Labour retaking City Hall, then the Goldsmith campaign will have to match Khan’s organisation, energy and enthusiasm.