British Prime Minister Theresa May has failed to strike a Brexit divorce deal with EU negotiators in Brussels despite earlier reports of a breakthrough agreement that would have kept British-ruled Northern Ireland aligned with EU regulations.

Both sides said they should unlock talks on future trade relations in the coming days.

"We will reconvene before the end of the week and I am also confident that we will conclude this positively," May said after a lunch on Monday with European Commission President Jean-Claude Juncker, who also voiced confidence in overcoming outstanding obstacles.

Met with PM @theresa_may. I was ready to present draft EU27 guidelines tomorrow for #Brexit talks on transition and future. But UK and Commission asked for more time. It is now getting very tight but agreement at December #EUCO is still possible. pic.twitter.com/oLQQHs9F8q — Donald Tusk (@eucopresident) December 4, 2017

He insisted the meeting, after days of intensive talks behind the scenes, was "not a failure" and credited the prime minister with hanging tough.

They spoke after government sources in Dublin said London had agreed to keep Northern Ireland "aligned" to EU regulations to avoid a "hard border" with the Irish Republic.

Word of that provoked an angry response from May's allies in Northern Ireland, demanding equal treatment with the rest of the UK.

Tthe idea of Northern Ireland remaining closely linked to the EU single market prompted speculation that, to avoid new barriers between Belfast and London, the British mainland would have to follow suit.

The leaders of Scotland and London, which voted against Brexit, demanded they be allowed the same EU relationship as Northern Ireland. Yet May has ruled out such differentiated treatment or staying in a customs union or the single market.

"She is a tough negotiator, and not an easy one," Juncker said in remarks reflecting concern in Brussels that concessions by May could fuel a disruptive push in London to unseat her.

"She is defending the point of view of Britain with all the energy we know she has," the EU chief executive told reporters.

"Despite our best efforts and the significant progress we and our teams have made over the past days on the remaining issues, it was not possible to reach a complete agreement."

"She's a tough negotiator - and not an easy one" EU Commission President Jean-Claude Juncker says of British Prime Minister Theresa May - "it was not possible to reach complete agreement today" #brexit @SBSNews — Brett Mason (@BrettMasonNews) December 4, 2017

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The sudden announcement that no deal had been reached swiftly squelched optimism that had spread earlier on Monday.

Irish government sources had said agreement had been reached on an overall deal for the Irish issues.

"The key phrase is a clear commitment to maintaining regulatory alignment in relation to the rules of the customs union and internal market which are required to support the Good Friday Agreement, the all-island economy and the border," one of the Irish government sources said.

The Irish border, which is now completely unguarded and barely even marked, would not change its physical appearance after Britain leaves the EU, Irish Foreign Minister Simon Coveney said. Fears that a "hard border" could disrupt the peace deal in the north had driven both sides to find solutions.