Negotiations are on a knife-edge after a hastily-arranged meeting on Sunday between EU chief negotiator Michel Barnier and Brexit Secretary Dominic Raab broke up without a breakthrough.
Discussions were said to have broken down after EU negotiators demanded a “backstop to the backstop” to prevent a return of a “hard border” between Northern Ireland and the Republic.
Theresa May has proposed the backstop - which would effectively keep Northern Ireland in the single market while a permanent solution is found - should apply to the whole of the UK.
However it is understood the EU is insisting it should be backed up by the original Northern Ireland-only backstop as it first proposed.
That could lead to customs checks on goods travelling between Northern Ireland and the rest of the UK - effectively imposing a “border in the Irish Sea” - something Mrs May has said is unacceptable.
The impasse threatens to throw into disarray carefully choreographed plans which would have seen EU leaders meeting in Brussels on Wednesday give the green light to a special summit in November to finalise the terms of the UK’s withdrawal from the bloc.
Following Sunday’s meeting in Brussels, Mr Barnier said that, despite “intense efforts”, there had been a failure to reach agreement on the border issue.
The Government said there were still “unresolved issues” relating to the backstop but that it remained committed to making progress at the European Council meeting.
But with Mrs May under siege from Tory Eurosceptics and her DUP parliamentary allies, her room for manoeuvre is severely restricted.
Writing in the Daily Telegraph, the DUP’s Westminster leader, Nigel Dodds, said: “One part of the UK cannot be left behind, bound to rules set in Brussels. The constitutional and economic consequences of such an approach would be catastrophic in the long run.”
For Labour, Sir Keir Starmer said the Government must now publish details of its revised proposals for the Irish border.
“At the moment we don’t even know what the proposal is that everybody has fallen out about over the weekend,” the shadow Brexit secretary told BBC Radio 4’s Today programme.
“That needs to be looked at and scrutinised in Parliament.”
Ireland’s ambassador to the UK, Adrian O’Neill, said events in Brussels were a “setback” and could increase the prospect of a no-deal Brexit.
He told BBC Radio 4’s Westminster Hour: “Time is running out, there is no doubt about that. In all member states, preparation for all eventualities are ramping up quite significantly.”
The special EU summit pencilled in for November to sign off a Brexit agreement could instead end up being used as an emergency meeting to discuss “no-deal” plans.
Talks will take place on Monday between DUP leader Arlene Foster and Taoiseach Leo Varadkar, while Mrs May will meet Sinn Fein leader Mary Lou McDonald in Westminster.
The febrile atmosphere in the Tory ranks has seen former Brexit secretary David Davis emerge as a potential successor to Mrs May.
Tory MP Nadine Dorries publicly suggested he could be the leader to deliver the kind of Brexit sought by Eurosceptics.
The Prime Minister’s critics believe the threshold of 48 Tory MPs calling for a no-confidence vote could be passed by Wednesday, depending on events in Brussels.
Allies of Mr Davis said he has been contacted by several Tory MPs urging him to run for the leadership and he is understood to be prepared to take part in a contest.
Ms Dorries backed Mr Davis for the top job, saying: “Getting May out and him becoming an interim leader may be the only way to deliver Brexit and FTA (a free-trade agreement).”