LONDON — The son of a London bus driver, Sadiq Khan has had a remarkable rise into the upper echelons of British politics. He grew up with seven siblings in a three-bedroom home in public housing and attended state schools before becoming a human rights lawyer and then a senior government minister.

Now Khan, 45, a lawmaker and former transport minister for the opposition Labour Party, is the favorite in the battle to become the next mayor of London. He would succeed Boris Johnson, the extroverted Conservative who has held the post since 2008 and is a leading figure in the campaign for Britain’s departure from the European Union.

A victory Thursday would make Khan the first Muslim to lead the city, where one in eight residents adheres to that faith, and when Britain is struggling to integrate minorities and combat radicalization.

Londoners have elected their mayor directly only since 2000, and just two politicians have held the post: Johnson and Ken Livingstone, who ran as an independent in 2000 and then for Labour in 2004.

Khan’s election would be a boost for the Labour Party at a time when Prime Minister David Cameron’s Conservatives are deeply divided on Europe. It would also probably strengthen Labour’s left-wing leader, Jeremy Corbyn, who is at odds with many of his own lawmakers, and whose leadership has been embroiled in a dispute about anti-Semitism in party ranks.

But the main policy battles being fought in London relate to the problems confronting a city that has an acute shortage of affordable housing and a creaking and overcrowded mass transit network. Khan’s closest rival is Zac Goldsmith, the Conservative candidate, who says he wants to make London “the greenest city in the world.’’

The two have battled over issues like transportation and how to build more homes to relieve sky-high real estate and rental costs. Khan points to the fact that the average rent in London is equivalent to 55 pounds, or about $80, a night — the price of a decent hotel in many European capitals. And Goldsmith has said the overcrowded conditions endured by London commuters “would be illegal for chickens, pigs, and cattle.’’

Appropriately, perhaps, for a globalized city that accommodates the extremes of wealth, Khan and Goldsmith are living examples of London’s diversity.

Goldsmith, 41, a son of the tycoon Sir James Goldsmith, inherited a fortune and was educated at Eton College, Britain’s most exclusive school. He edited the magazine The Ecologist, founded by his uncle, and was elected to Parliament in 2010.

The fight between the two main contenders has not been pretty, bringing delicate issues of ethnicity and religion into a mayoral election in a way never seen here before.

Goldsmith has called Khan dangerous and without principle and said it was his job “to prevent London falling into the wrong hands.’’

Khan has suggested that Goldsmith has run a desperate and divisive campaign, has targeted voters on religious or ethnic lines, and has no track record suggesting he could perform as London mayor.

Goldsmith has also accused Khan of giving tacit support to extremists by speaking on the same platform in the past as those who espoused radical views or who had been accused of supporting terrorists, when he was a human rights lawyer and campaigner.

Part of Khan’s pitch to Londoners is that, as an observant Muslim (he does not drink), he has a plan to tackle extremism and that like many people in London he has “multiple identities.’’

“I’m a Londoner, I’m European, I’m British, I’m English, I’m of Islamic faith, of Asian origin, of Pakistani heritage, a dad, a husband,’’ Khan said in an interview, before accusing his main opponent of seeking to “divide communities.’’

Khan routinely highlights his humble origins, but he has been careful not to attack Goldsmith over his wealth.

“I don’t hold his background against him,’’ Khan said. “None of us are responsible for who our parents are — our families — that’s not fair.’’

The personalization of the campaign underlines the lack of deep policy differences confronting Londoners, analysts said.

Neither candidate has challenged London’s existing model of openness to immigration and investment, allied to rapid economic growth, said Tony Travers, the director of a research group on London at the London School of Economics and Political Science.

In policy terms, “it is hard to argue that they are radically different from each other,’’ he said.