By Elizabeth Piper and William James

LONDON (Reuters) - Sadiq Khan, a Muslim opposition lawmaker, is on course to be elected London's mayor on Thursday, loosening the ruling Conservatives' hold on Britain's financial center after a campaign marred by charges of anti-Semitism and extremism.

The fight to run the British capital has pitted the Labour Party's Khan, 45, the son of a bus driver who grew up in public housing, against Conservative Zac Goldsmith, 41, the elite-educated son of a billionaire financier.

But rather than their social backgrounds, it has been accusations of smears over Khan's Muslim faith and anti-Semitism in the Labour Party that have dominated the campaign to replace Conservative Boris Johnson as mayor of the city of 8.6 million people which is usually known for its tolerance.

Mayors of London, which is home to the City financial district, are responsible for areas such as policing, transport, housing and the environment.

London is the top prize in local elections being held across Scotland, Wales and northern England, and there was little let-up in the accusations against Khan, who has a big lead in opinion polls, as campaigning ran into its final hours.

During a heated parliamentary debate on Wednesday, British Prime Minister David Cameron accused Khan of sharing "a platform with an extremist who called for Jews to drown in the ocean".

Labour Party leader Jeremy Corbyn in turn accused the Conservatives of "smearing" Khan. He said one of the men Cameron had accused Khan of sharing a platform with had also been close to Goldsmith.

Khan, who would be London's first Muslim mayor, says he has fought extremism all his life and that he regrets sharing a stage with speakers who held "abhorrent" views.

ANTI-SEMITISM ROW

The former human rights lawyer has also had to distance himself from Corbyn after a row over anti-Semitism.

The Labour leader ordered an inquiry into charges of anti-Semitism after suspending Ken Livingstone, a political ally and a former London mayor, for saying Adolf Hitler had supported Zionism.

Khan was quick to condemn the comments and on Tuesday told reporters in his final stump speech of the campaign at a venue overlooking the River Thames: "I've had lots of people contact me from the Jewish faith sending me their support".

The row has failed to dent Khan's lead in opinion polls, a situation the Labour Party would like to be replicated in Thursday's other local and regional elections.

But in Corbyn's first electoral test since taking over the party last September, analysts say Labour could lose dozens of seats in some of its traditional strongholds.

After Corbyn expressed confidence that Labour would gain seats, his spokesman qualified his remarks on Thursday, saying he rather wanted to say: "We're not in the business of losing seats and we'll be fighting to win as many as possible tomorrow."

(writing by Elizabeth Piper, editing by Gareth Jones)