British papers said at least two or three ministers could quit over May's concessions on terms for keeping the Irish border open after Britain leaves the European Union on March 29.
While there are reportedly no more talks to be held between Number 10 and Brussels before Wednesday, Downing Street said the Government is still "committed to making progress".
But despite suggestions that negotiators had come to a tentative agreement, news broke that key issues including the Irish border question remained unresolved.
Hopes were raised that a divorce settlement was close yesterday when Brexit Secretary Dominic Raab made an unscheduled trip to Brussels for talks with Michel Barnier.
The issue of the border between Northern Ireland and the Republic of Ireland, which will become the UK's border with the European Union, is one of the last remaining obstacles to achieving a divorce deal with Brussels.
At least one Cabinet minister is considering resigning over the prime minister's Brexit position, according to people familiar with the situation.
Friends of the former Brexit secretary told the paper that fellow leadership hopeful, Boris Johnson, is struggling to build a large enough base of MPs prepared to put him on the final ballot sent out to members.
Neither side is sure they will clinch a deal in time to meet a self-imposed Monday deadline before a meeting of leaders in Brussels on Wednesday.
The Democratic Unionist Party (DUP) which props up May's minority government is also putting huge pressure on May, with its leader Arlene Foster suggesting that she is "ready" to force a no-deal Brexit rather than accept the provisional deal agreed by United Kingdom negotiators.
Wednesday's summit will be the first time European Union leaders have come together since September, when UK Prime Minister Theresa May's efforts to reach a Brexit deal was rejected.
"Adopting a constructive and transparent approach would be in the national interest", says the letter, organised by the Economists for Free Trade group of Eurosceptic economists and former Brexit minister Steve Baker.
One of the issues that has been hampering progression of a deal has been that of the Irish border and a potential backstop - a mechanism to avoid a hard border with Northern Ireland.
THERESA May's allies have moved to reassure the DUP that is she is still "a very Unionist Prime Minister".
"This is one of the most fundamental decisions that government has taken in modern times".
David Davis, who quit as Brexit secretary in July over May's broad blueprint, wrote in The Sunday Times newspaper her plans were "completely unacceptable" and urged ministers to "exert their collective authority" this week.
Health minister Matt Hancock suggested the backstop could be temporary without such a date, an argument that may fall flat for some eurosceptic lawmakers who are calling for May to "chuck Chequers", her Brexit plan named after her country residence.
Eurosceptics want a time limit for how long Britain will keep following the EU's customs rules before it can strike out on its own and sign independent trade agreements with new partners.
DUP leader Arlene Foster is said to have left a meeting with Mr Barnier last week convinced that no deal was the "most likely outcome".
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