Mr Khan said the referendum should offer voters the choice of staying in the EU against any deal the government manages to strike – or against a “no-deal” Brexit if an agreement cannot not reached.
“I’ve become increasingly alarmed as the chaotic approach to the negotiations has become mired in confusion and deadlock, leading us down a path that could be hugely damaging – not only to London, but the whole country,” said Mr Khan.
He said the government had failed to put the national interest ahead of party politics and that only a “vague” bad deal or a no-deal were possible.
“Both these scenarios are a million miles away from what was promised during the EU referendum campaign,” said London’s mayor, claiming that independent analysis had forecast 500,000 fewer jobs across Britain by 2030 if a Brexit deal is not reached.
Mr Khan added: “The need for another public vote on Brexit was never inevitable, or something I ever thought I’d have to call for.
“But the reality is that the abject failure of the government – and the huge risk we now face of either a bad deal or a ‘no deal’ Brexit – means that giving people a fresh say on our future is now the right, and only, approach left for the good of our country.”
Prime Minister Theresa May has said that “giving in” to calls for a second referendum on the final terms of the UK’s withdrawal from the EU would be “a gross betrayal of our democracy”.
She has repeatedly dismissed calls from the People’s Vote, a cross-party group that includes several high-profile figures and MPs, for a second Brexit referendum.
Labour leader Jeremy Corbyn has said he is not backing calls for another vote – but Mr Khan’s move is likely to add pressure on him to reconsider.
Mrs May is facing strong opposition to her Chequers blueprint for leaving the EU from hardline Tory Brexiteers.
Without support from at least some opposition MPs, the Government is likely to struggle to get any agreement through Parliament.
A loosely-worded deal could be struck that would be able to be sold to all sides.
Shadow Brexit secretary Sir Keir Starmer, however, said Labour would vote down any attempts to force the country into a “blind Brexit”.
In a letter to Brexit Secretary Dominic Raab published by The Sunday Times, he said the government must meet its key Brexit tests, which include delivering the “exact same benefits” as the UK currently has as members of the single market and customs union, to win Labour support.
Sir Keir added: “But, let me be equally clear: a vague political declaration would not meet those tests. Labour will not – and cannot – vote for a blind Brexit.”