Theresa May pulled out of a possible deal to break the Brexit logjam at the last moment after meeting fierce resistance from Unionists to proposals which would align Northern Ireland's regulations with the Republic, Ireland's prime minister has claimed.

Crunch talks in Brussels between Mrs May and European Commission president Jean Claude-Juncker ended without a deal, after the PM broke off from negotiations for urgent telephone talks with Democratic Unionist Party leader Arlene Foster.

Discussions are set to resume later this week, with both Mrs May and Mr Juncker declaring themselves "confident" that a solution can be found in time for a key summit of the European Council on December 14.

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Mr Juncker said that "significant progress" had been made, but it was not possible to reach a "complete agreement" on Monday, while Mrs May said it was clear that both sides wanted to "move forward together".

But Taoiseach Leo Varadkar later said that he was informed that agreement had been reached on the key issue of the Irish border before the dramatic intervention of the DUP.

Reports that Mrs May was on the verge of agreeing a deal on "regulatory alignment" between Northern Ireland the Republic led the DUP to warn it would oppose the deal if it meant Northern Ireland being subjected to different rules from the rest of the UK.

Video: DUP vows to block any move that "separates" Northern Ireland from the rest of the UK

The DUP, which props up Mrs May's minority Government in the House of Commons, has previously warned it could withdraw its support in Westminster if a deal is proposed which threatens the integrity of the United Kingdom.

Speaking at Stormont, DUP leader Arlene Foster said: "We have been very clear. Northern Ireland must leave the EU on the same terms as the rest of the United Kingdom. We will not accept any form of regulatory divergence which separates Northern Ireland economically or politically from the rest of the United Kingdom."

In a press conference in Dublin, Mr Varadkar said he was "surprised and disappointed" by the break-up of talks, which came after representatives of the UK Government and EU chief Brexit negotiator Michel Barnier informed Irish negotiators that a form of words had been found which might satisfy the Republic's demand for a "cast-iron guarantee" that there would be no hard border with Northern Ireland after Brexit.

"The Irish negotiating team received confirmation from the British Goverment and the Barnier taskforce that the United Kingdom had agreed a text on the border that met our concerns," said Mr Varadkar.

"This text would form a part of the broader EU/UK agreement on phase one (of the Brexit negotiations) and allow us all to move on to phase two."

Mr Varadkar said that he had confirmed Ireland's agreement to the text to both Mr Juncker and European Council president Donald Tusk.

"I am surprised and disappointed that the British Government now appears not to be in a position to conclude what was agreed earlier today," he said. "I accept that the Prime Minister has asked for more time, and I know that she faces many challenges and I acknowledge that she is negotiating in good faith.

"But my position and that of the Irish Government is unequivocal and is supported by all the parties in Dail Eirann and I believe the majority of people on these islands. Ireland wants to proceed to phase two - It's very much in our interests to do so. However we cannot agree to do this unless we have firm guarantees that there will not be a hard border in Ireland under any circumstances."

Since the break-up of talks, Mr Juncker had confirmed that "Ireland's position remains Europe's position", said Mr Varadkar, adding: "I still hope this matter can be concluded in the coming days, as was agreed."

In a brief appearance before the cameras after their meeting, both Mrs May and Mr Juncker insisted that their talks - which extended from lunch into the late afternoon - had been "constructive".

Mr Juncker said: "We now have a common understanding on most relevant issues, with just two or three open for discussion.

"These will require further consultation, further negotiation and further discussions ... but I have to say that we were narrowing our positions to a huge extent."

And Mrs May said: "We have been negotiating hard. And a lot of progress has been made. And on many of the issues there is a common understanding.

"And it is clear, crucially, that we want to move forward together. But on a couple of issues some differences do remain which require further negotiation and consultation."

Insisting he remained "confident" that agreement would be reached in time for the leaders of the remaining 27 EU states to give a green light for the start of trade talks at next week's summit, Mr Juncker said: "This is not a failure, this is the start of the very last round."

Labour leader Jeremy Corbyn said: "The real reason for today's failure is the grubby deal the Government did with the DUP after the election.

"Each passing day provides further evidence that Theresa May's Government is completely ill-equipped to negotiate a successful Brexit deal for our country."

READ MORE: Theresa May’s efforts in Brexit talks a shambles, Commons told

In a message on Twitter following a meeting with Mrs May, Mr Tusk made clear that he had been preparing to move negotiations on to the second phase until the last-minute call for more time.

"I was ready to present draft EU 27 guidelines tomorrow for Brexit talks on transition and future. But UK and Commission asked for more time," said Mr Tusk.

"It is now getting very tight but agreement at December European Council is still possible."

Maintaining a soft Irish border has emerged as the key sticking point in Brexit negotiations, after London indicated it was ready to up its offer on the so-called "divorce bill" to as much as £50 billion.

Regulatory alignment could mean both Ireland and Northern Ireland following the same rules governing trade, to ensure that goods can continue to move freely across a "soft" border with no checks.

But critics say that it would effectively move the customs border between the UK and the Republic into the Irish Sea.

Mrs Foster spoke out after Ireland's deputy premier Simon Coveney said it appeared the Dublin Government's concerns were set to be addressed fully, with a post-Brexit border which would be "invisible" with "no barriers".

Leaders of devolved governments in Scotland, Wales and London added a further complication by announcing that if Northern Ireland was to be offered special status after Brexit, other parts of the UK should be offered a similar opportunity.

Scottish First Minister Nicola Sturgeon, who opposes Brexit, said: "If one part of UK can retain regulatory alignment with EU and effectively stay in the single market - which is the right solution for Northern Ireland - there is surely no good practical reason why others can't."

And Welsh First Minister Carwyn Jones said: "We cannot allow different parts of the UK to be more favourably treated than others.

"If one part of the UK is granted continued participation in the single market and customs union, then we fully expect to be made the same offer."

READ MORE: Independence will become "increasingly irresistible" if Scotland taken out of EU single market

London's Labour mayor Sadiq Khan said the deal reportedly being discussed in Brussels would have "huge ramifications for London", which voted overwhelmingly to remain in the EU.

"If Theresa May has conceded that it's possible for part of the UK to remain within the single market and customs union after Brexit ... and a similar deal here could protect tens of thousands of jobs," said Mr Khan.

Conservative MPs and peers who were briefed in Westminster by Brexit minister Steve Baker and Mrs May's chief of staff Gavin Barwell said they had been given the impression that the PM had not agreed to Irish proposals on keeping Northern Irish regulations in line with the south.

Leading Brexiteer Jacob Rees-Mogg told reporters: "I don't think that can possibly happen. The Government doesn't have a majority for that."

And backbench Remain supporter Anna Soubry said that the "simple solution" would be for the whole of the UK to remain in the single market and customs union.

"Nobody could want one part of our country to have a different set of rules to another part of our country," said Ms Soubry. "On that, everybody is agreed."

Mr Rees-Mogg said that Dublin was "promoting the creation of a united Ireland" by trying to force Britain's hand on the border. But Mr Varadkar insisted there was "no hidden agenda" and that his policy was driven by the terms of the Good Friday Agreement which sealed the peace process.