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Writing in the Observer, the Labour politician said that with the UK due to leave the EU in six months, it now faced either a "bad deal" or "no deal".
The debate had also become "more about Boris Johnson's political ambitions" than what was good for the UK, he said.
Tory minister Michael Gove said Mr Khan wanted to frustrate the will of voters.
The mayor's comments are also at odds with the Labour's official policy, which is to respect the outcome of the referendum but "leave all options on the table" if a deal is not agreed by Parliament.
Former Tooting MP Mr Khan said that although he campaigned to remain in the EU, he had accepted "the will of the British people was to leave the EU".
He said he had never expected to back calls for a second referendum, but had become "increasingly alarmed as the chaotic approach to the negotiations had become mired in confusion and deadlock".
With time running out for the British government to negotiate a final deal with the EU before March 2019, Mr Khan said the UK was left with two "incredibly risky" possibilities.
"Both these scenarios are a million miles from what was promised during the referendum campaign, only further exposing the lies and mistruths sold to the public," he wrote in the paper.
"I don't believe Theresa May has the mandate to gamble so flagrantly with the British economy and people's livelihoods."
'Not a rerun'
Appearing on BBC One's Andrew Marr Show, Mr Khan was shown comments he made to LBC radio after the Brexit referendum, in which he warned a fresh poll would lead to "even more cynicism" among voters.
He told Marr: "It's really important that this is not a re-run of the referendum but the British public having a say for the first time on the outcome."
Pushed on what voters should be asked in a fresh vote, he said it should be a choice between the government's deal and remaining in the EU.
Asked whether that would disenfranchise supporters of a hard Brexit, Mr Khan added: "My point is this. Rather than having a bad deal or a no deal, let's put that to the British public with the option of staying in the EU." [...]