In a column for the Observer newspaper published late Saturday, Khan said that time was "rapidly running out" while the government of Prime Minister Theresa May looked "unprepared and out of its depth."
Failure to reach a deal with Brussels, or a bad deal, could result in the loss of 500,000 jobs and result in understaffed hospitals, businesses suffering and the police having to prepare for civil unrest, he warned.
"I don’t believe May has the mandate to gamble so flagrantly with the economy and people’s livelihoods," said Khan, a member of the Labour opposition party who campaigned to remain in the EU head of the 2016 referendum.
Khan has no direct role to play in the Brexit process, but he is a very influential member of the political class.
He said Britons should be given a vote on any deal, or a vote on a no-deal alongside the option of staying in the EU.
Britain is set to leave the EU on March 29, but May has struggled to come up with an exit deal palatable to Brussels, or even to her own Conservative party.
Her former foreign secretary Boris Johnson, who is regularly rumoured to be plotting to oust her, has also come out against her proposals, known as the Chequers agreement.
"It’s time to take this crucial issue out of the hands of the politicians and return it to the people," Khan wrote.
"The government’s abject failure – and the huge risk we face of a bad deal or a 'no deal' Brexit – means that giving people a fresh say is now the right – and only – approach left for our country."