Brexit talks broke up in Brussels without a deal, after a proposed solution for the Irish border met fierce resistance from the Democratic Unionist Party.
Discussions are set to resume later this week, with both Prime Minister Theresa May and European Commission president Jean-Claude Juncker declaring themselves "confident" that a solution can be found in time for a key summit of the European Council on December 14.
Mrs May began her lunchtime meeting with the Commission president with hopes high that Brussels would be able to declare that sufficient progress had been made on the so-called "divorce issues" in order to allow the leaders of the remaining 27 EU nations to give the green light for trade talks to begin next week.
But 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 not back any agreement which threatened the territorial integrity of the UK.
Mr Juncker said the meeting was "friendly and constructive".
He went on: "I have to say that she's a tough negotiator, and not an easy one, and she's defending the point of view of Britain with all the energy we know she has, and this is the same on the side of the European Union.
"Despite our best efforts and significant progress we and our teams have made over the past days on the three main withdrawal issues, it was not possible to reach a complete agreement today.
"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.
"We stand ready to resume the negotiations with the United Kingdom here in Brussels later this week.
"But I have to say that we were narrowing our positions to a huge extent today, thanks to the British Prime Minister, thanks to the willingness of the European Commission to have a fair deal with Britain.
"I'm still confident that we can reach sufficient progress before the European Council of December 15.
"This is not a failure, this is the start of the very last round. I'm very confident that we will reach an agreement in the course of this week."