As negotiations continued in Brussels ahead of a summit starting on Wednesday, Cabinet Ministers were urged to speak out against her plans and potentially quit in protest at any deal which could see the UK indefinitely tied in to the EU's customs union.
Former Brexit Secretary David Davis said the plan was "completely unacceptable" and urged Cabinet Ministers to "exert their collective authority".
Tory MP Nadine Dorries suggested that Mr Davis should be installed as interim leader, claiming that was the only way to secure the kind of free-trade deal Brexit demanded by Eurosceptics.
Cabinet Minister Matt Hancock sought to play down speculation that some of his colleagues might quit over the Brexit plans, but was unable to say whether a fixed deadline for any customs arrangement would be written into a deal with Brussels.
He told the BBC's Andrew Marr Show: "There are different ways that you can make sure that something is credibly time limited and that's what I want to see."
Tory vice chairman James Cleverly told Sky News' Sophy Ridge on Sunday that Cabinet ministers should use their position to influence Brexit policy rather than resign.
"They should use that position to put their ideas forward, I would suggest that's a much more constructive and helpful way of influencing the direction of the negotiations with the EU," he said.
Writing in the Sunday Times, Mr Davis, who quit in July over Brexit, said: "It is time for the Cabinet to exert their collective authority. This week the authority of our constitution is on the line."
The newspaper said at least nine ministers want Mrs May to change course when the Cabinet meets on Tuesday.
Speculation about possible resignations has centred on Andrea Leadsom, Penny Mordaunt and Esther McVey, but the report also indicated that Scottish Secretary David Mundell and Scottish Tory leader Ruth Davidson could quit because of the sensitivity of the issue in relation to calls for Scottish independence.
Mrs May's own position also appeared in jeopardy, with as many as 44 letters demanding a vote of no confidence reportedly submitted to the Conservative 1922 Committee - just four short of the number required to trigger a ballot.
Ms Dorries said that Mr Davis had always focused on changing the policy, rather than the Prime Minister.
But she added: "Getting May out and him becoming an interim leader may be the only way to deliver Brexit and FTA (a free-trade agreement)."
Dozens of Tory Eurosceptics have also written to Chancellor Philip Hammond demanding the release of the analysis behind the cross-Whitehall Brexit impact assessments, amid claims they were "Project Fear"-style scaremongering.
The febrile atmosphere in Tory ranks comes as negotiators in Brussels attempt to reach a deal on the withdrawal agreement for the UK's exit from the EU.
The issue of the border between Northern Ireland and Ireland is one of the last remaining obstacles to achieving a divorce deal, with wrangling continuing over the nature of a "backstop" to keep the frontier open if a wider UK-EU trade arrangement cannot resolve it.
The European Union's version of the backstop, which would see just Northern Ireland remain aligned with Brussels' rules, has been called unacceptable by Mrs May and is loathed by the DUP.
Mrs May's counter-proposal is for a "temporary customs arrangement" for the whole UK, but Tory Brexiteers fear this becoming an open-ended position which would prevent free trade deals with countries around the world.
The Northern Irish situation is a particular headache for Mrs May because her minority administration depends on the votes of the 10 DUP MPs.
According to a private email exchange between senior UK officials, seen by the Observer, DUP leader Arlene Foster has indicated her party was "ready for a no-deal scenario, which she now believed was the likeliest one".
Mrs May's hopes of getting a Brexit deal through Parliament could depend on the actions of Labour MPs, with former Minister Caroline Flint telling Sky she would back a "reasonable" deal.
The Don Valley MP said: "I think if a reasonable deal is on the table the question for some of my Labour colleagues is 'why wouldn't you support a deal, why would you stand along (with) Boris Johnson or Jacob Rees-Mogg who want us to crash out without a deal?"
But Shadow Foreign Secretary Emily Thornberry told the BBC: "We said we wanted a meaningful vote and we can't see why we should have, on the one hand Theresa May's nonsense and on the other hand a no deal, because that's what they're threatening us with.
"If she comes back with something that's just a fudge she's cooked up with Brussels... we're not voting for something that's essentially a bridge to nowhere."