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.
Prime Minister Theresa May’s official spokesman said: “We have made real progress in a number of key areas. However, there remain a number of unresolved issues relating to the backstop.
“The European Union and the UK are both clear that they want to secure a good deal and that is what both sides are working towards.
“We remain confident of getting a deal because it is in the interests of both the UK and the European Union.
“We’ve said that we want to continue to make progress in the coming days and weeks. That’s what we are focused on.”
He added: “The EU continues to insist on the possibility of a customs border down the Irish Sea. This is something which Parliament has already unanimously rejected and is not acceptable to the Prime Minister.”
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.”
Arriving in Brussels for an EU Foreign Affairs Council meeting, Irish Tanaiste Simon Coveney said Dublin and the EU simply wanted Theresa May to follow through with backstop agreements already made in March and December.
He told reporters “a backstop can’t be time-limited”, adding: “The backstop will be there unless and until something else is agreed, but unless you have something to replace it well then the backstop needs to be there as an insurance mechanism.
“That is all we are asking for, that’s all the Michel Barnier taskforce is also looking for now in terms of legal text.”
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).”