Khan said Britain now faces either a bad deal or a no-deal Brexit, both of which are "incredibly risky" for Britain.
Khan denied claims he had called for the vote in order to distract from his record as London mayor, saying a record number of homes had been started in the capital past year.
He says the prime minister's handling of Brexit negotiations has become "mired in confusion and deadlock" and was leading the country down a damaging path. Britain is due to leave the bloc next March.
But Labour frontbencher Barry Gardiner yesterday told Sky's Sophy Ridge a fresh referendum would give Theresa May a "lifeline".
"If this government can not do what it is supposed to and govern, then we need actually to change the government", he told Sky News.
May, who has faced mounting criticism at home over her handling of the Brexit negotiations, has repeatedly pushed back against suggestions of holding a second public vote on the issue.
Theresa May said on Sunday she was focused on securing a deal based on her Chequers plan, but admitted to being "a little bit irritated" by constant speculation about her position.
"So what I'm saying to the government is for the first time, the British public should have a say on the outcome of those negotiations, including the option of staying in the European Union".
"This debate is not about my future".
"I don't want manufacturers to feel that they've got to operate under all sorts of different rules, because that complicates life for them and that potentially means business leaving this country", she said. Finance minister Philip Hammond said the government should heed the warning from the International Monetary Fund.
British Prime Minister Theresa May has told Tory MPs that the alternative to her potential Brexit deal with the European Union is no deal.
Appearing on the BBC's Andrew Marr Show, where he was played a voice clip in which he argued against the prospect of a second poll, asserting it would fuel massive "cynicism" amongst voters, Khan argued another vote would not be "a re-run of the referendum" but the public having their say over the outcome of negotiations.