Labour demands that Britain retain "the exact same" perks it now has within the EU's customs union and single market - something May's so-called Chequers plan does not meet and which the EU rules out since London chose to leave both.
Negotiations were stepped up this morning after UK Brexit secretary Dominic Raab made an unexpected visit to see the EU's negotiator Michel Barnier.
Diplomatic sources in Brussels confirmed to AFP that negotiations have been suspended until Wednesday, when the leaders of the other 27 European Union member states will be in town on the eve of their October summit, which had already been billed as a "moment of truth".
European Union negotiators have criticized that proposal, and said on Sunday it was clear that, as things stood, May did not feel she could get a deal through her cabinet, which meets on Tuesday.
Raab's predecessor, David Davis, wrote in the Sunday Times that May's plans for continued close economic ties with the European Union even after Britain leaves the bloc is "completely unacceptable" and must be stopped by her ministers.
Former Brexit secretary David Davis has urged a Cabinet revolt against the Prime Minister's approach to the talks.
Eurosceptic lawmakers in her Conservative Party are also concerned about a backstop, fearing it could keep Britain in the EU's sphere of influence possibly forever and stop the country from striking new trade deals with the rest of the world.
The issue of the Northern Irish backstop - a mechanism to avoid a hard border - is at the heart of the troubles facing Mrs May's leadership.
There will be no further attempt to resolve the impasse before the European Union summit in the Belgian capital on Wednesday, according to people familiar with the matter, and officials on both sides are concerned time is running out to get an agreement before the U.K.'s exit in March.
Keeping an open border between Northern Ireland and the Republic of Ireland could force the rebuilding of customs and law enforcement posts.
The issue of the border between Northern Ireland and Ireland is one of the last remaining obstacles to achieving a divorce deal.
Theresa May's Brexit plans have reached a crisis after she dramatically pulled the plug on a fledgling divorce settlement over the EU's demands for the Irish backstop.
Her spokeswoman said yesterday London would present further proposals, which she described as "regulatory aspects", for the backstop border arrangement "in due course".
Under pressure from all sides, Theresa May told journalists at Downing Street reception that talks on the key issue of the Irish border were likely to continue until November, while cabinet ministers who met with May yesterday evening were cited by the Financial Times as saying the border issue was close to being settled.
The DUP are staunchly pro-UK, pro-Brexit and fiercely opposed to any moves that could put distance between Northern Ireland and the British mainland.
Mrs Foster - whose ten DUP MPs wield a massive amount of political power as they prop the Tories up in No10 - said: 'Great Britain is our largest market by far and we can not have barriers'.
Health Secretary Matt Hancock sought to play down the potential for Cabinet resignations but was unable to say whether a fixed deadline for any customs arrangement would be written into a deal with Brussels.
Speaking to MPs in the Commons on Monday, the Prime Minister said there are still issues surrounding the Irish border question. DUP leader Arlene Foster said Thursday that May and her government "could not in good conscience recommend a deal which places a trade barrier on United Kingdom businesses moving goods from one part of the Kingdom to another".