Theresa May is engaged in a race against the clock to find the key to break deadlock in Brexit talks.
The Prime Minister's hopes of securing agreement on the terms of Britain's EU withdrawal were dashed on Monday when the Democratic Unionist Party refused to accept proposals which would have shifted Northern Ireland's customs border to the Irish Sea.
Mrs May is planning to return to Brussels before the end of the week, with time running out to persuade leaders of the remaining 27 EU nations at a summit on December 14-15 that "sufficient progress" has been made on divorce issues to move Brexit negotiations on to their second phase.
This will deal with trade and the transition to a new relationship.
On Tuesday she was expected to speak by phone with DUP leader Arlene Foster as she grapples to find a form of words acceptable to the Northern Irish party, on which she relies to prop up her minority administration at Westminster.
Mrs May had to break off from talks with European Commission president Jean-Claude Juncker on Monday for an urgent call with the DUP leader, after she dramatically declared her party's implacable opposition to proposals which would have imposed "regulatory alignment" between Northern Ireland and the Republic in order to avoid the need for a hard border.
Irish premier Leo Varadkar said that the deal had been agreed by the European Commission, UK and the Republic before the process was thrown into disarray by Mrs Foster's eleventh-hour intervention. He said he was "surprised and disappointed" by Mrs May's request for more time.
European Council president Donald Tusk confirmed that, until the surprise developments at Stormont, he had been preparing to issue new negotiating guidelines for the second phase of talks on Tuesday.
It is also understood that differences also remain between the EU and Britain over the issue of European Court of Justice jurisdiction over EU nationals living in the UK after Brexit.
Mrs May insisted she was still "confident" of getting a green light for trade talks at next week's summit.
Failure to do so would risk throwing the whole Brexit process into crisis, as many companies are believed to be preparing to activate contingency plans to start moving staff and activities out of the UK if there are no signs of progress by Christmas.
EU officials are understood to believe that a text of the deal must be thrashed out by the end of the week to allow it to be included in draft summit conclusions and give other leaders time to consult their own governments - and in some cases parliaments - before convening in Brussels.
Prominent Tory MPs voiced their opposition to any deal which threatened the territorial integrity of the United Kingdom by forcing Northern Ireland to operate under different regulations from the mainland.
Leading Brexiteer Jacob Rees-Mogg said: "I don't think that can possibly happen. The Government doesn't have a majority for that."
Leaders of devolved administrations in Scotland, Wales and London that they would want the option to adopt in their parts of the UK any special status afforded to Northern Ireland.
Labour leader Jeremy Corbyn said the failure of talks showed Mrs May's Government was "completely ill-equipped to negotiate a successful Brexit deal for our country", while former Ukip leader Nigel Farage said she should "leave office now".
But many Conservatives were optimistic of a deal being struck.
The chairman of the Commons Northern Ireland Committee, Tory MP Andrew Murrison, told BBC2's Newsnight: "We understand that later this week there is every prospect of a deal satisfying the European Union that, in its words, sufficient progress has been made which will set us up for a favourable conclusion to the summit."