Theresa May is expected to speak by phone with the DUP leader Arlene Foster on Tuesday in a bid to find a solution to the Brexit talks deadlock.
The Prime Minister's hopes of securing an agreement on the terms of the UK's EU withdrawal were dashed on Monday, when the DUP 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.
Mrs May had to break off from talks with European Commission president Jean-Claude Juncker on Monday for an urgent phone call with the DUP leader, after she dramatically declared her party's opposition to proposals which would have imposed "regulatory alignment" between Northern Ireland and the Republic of Ireland in order to avoid the need for a hard border.
Speaking at Stormont Mrs 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."
EU Commission President Jean Claude Juncker and Theresa May addressed a press conference where they said "significant progress had been made" but that it had "not been possible to reach a complete agreement".
Taoiseach Leo Varadkar said that the deal had earlier been agreed by the European Commission, UK and the Republic before the process was stalled by Mrs Foster's intervention.
He said he was "surprised and disappointed" by Mrs May's request for more time.
Downing Street insist it was not only the DUP's response that prevented a deal, but that there remain outstanding concerns on citizens' rights and the financial settlement.
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 next week.