The Daily Telegraph newspaper said Northern Ireland's Democratic Unionist Party (DUP), whose 10 MPs help May command a slim majority, would vote against the deal.
Their alliance with the Tories is over unless the prime minister is replaced, the paper said, citing sources close to DUP leader Arlene Foster.
The prime minister admitted "concerns about the backstop" solution to the Irish border question within the deal, which Brexit supporters fear would keep Britain tied indefinitely into a customs union.
Critics also believe May has conceded too much to Brussels in other key areas, while EU supporters are calling for a second referendum on a final deal.
May, however, said there would be no second vote "as far as I'm concerned".
The 585-page draft aims to ensure a smooth divorce from the EU after more than four decades of membership and outlines a transition period for both sides to adjust to the break.
Key provisions seek to avoid a hard border between EU member Ireland and the British province of Northern Ireland, protect citizens' rights and settle Britain's last bill.
Amid the political turmoil, the pound dropped by 2 per cent against the dollar to a one-month low and a similar amount against the euro.
Tory MP Jacob Rees-Mogg, leader of the Brexit hardline European Research Group, submitted a letter of no-confidence in the prime minister saying: "It would be in the interest of the party and the country if she were to stand aside."
At least 48 letters from Conservative MPs are required to trigger a vote of no-confidence in the party leader, but a majority of the party's 315 lawmakers would have to vote against May in order for her to be ousted.
Although other MPs have already sent letters, all eyes were on Rees-Mogg given his influence over Brexit supporting MPs.
The MP told reporters that a challenge could be launched within weeks.
But veteran MP Kenneth Clarke, an arch-europhile, told Sky News that May would win any confidence vote, saying "there isn't an alternative".
EU leaders will hold an extraordinary Brexit summit on November 25.
If they approve the agreement, the British parliament is scheduled to vote on it in early December.
Raab said there would be a devastating impact on public trust in the government unless it changed course on Brexit.
"I cannot reconcile the terms of the proposed deal with the promises we made to the country in our manifesto," he said in his resignation letter.
Brexit hardliner Esther McVey also quit her work and pensions secretary post.
Suella Braverman resigned as a junior Brexit minister and Shailesh Vara quit as a junior Northern Ireland minister.
In parliament, Jeremy Corbyn, leader of the main opposition Labour Party, told May: "The government must now withdraw this half-baked deal.
"This is not the deal the country was promised." Vince Cable, leader of the Liberal Democrats, later led a rally in Parliament Square over the "botched deal", with protesters carrying placards reading "Brexit is failing" and "We need a people's vote".
"It's total chaos. She's never going to get anything through parliament. The whole house of cards is collapsing," said writer Emma Roper-Evans, 53.
May had secured her cabinet's "collective" approval for the agreement during a stormy five-hour meeting on Wednesday and European leaders hailed the tentative deal.
German Chancellor Angela Merkel said she was "very happy" that the EU and Britain had reached a draft agreement but French Prime Minister Edouard Philippe warned the prospect of Britain crashing out without a deal was "still on the table".
In Brussels, EU President Donald Tusk said member states would have until Tuesday to examine the deal and to agree the wording of a parallel political statement setting out goals for the bloc's future relations with London.
The agreement was also welcomed along the Irish border.
"If Theresa May has got any sort of a deal I think it's a miracle," said businessman Patrick Hughes, owner of an animal feed business in the border village of Jonesborough.
"I think she was fed to the lions a bit," he said.
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