* May met Juncker and Barnier over lunch
* Talks to go on, deal seen in coming days
* Irish deal offers breakthrough to trade talks
* Scots and London immediately seek own customs union (Updates with no deal agreed yet)
BRUSSELS, Dec 4 (Reuters) - British Prime Minister Theresa May failed to strike a Brexit divorce deal with EU negotiators in Brussels on Monday 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 with European Commission President Jean-Claude Juncker, who also voiced confidence in overcoming outstanding obstacles.
He insisted the meeting, after days of intensive talks behind the scenes, was "not a failure" and, well aware the EU does not want to humiliate May as she accepts many of its terms for withdrawal, 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 sent the pound higher on hopes of rapid trade talks but also provoked an angry response from May's allies in Northern Ireland, demanding equal treatment with the rest of the United Kingdom.
Underlining the conundrums of Brexit, the 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."
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.
Juncker spoke to Irish Prime Minister Leo Varadkar by phone. EU lawmaker Philippe Lamberts, who met Juncker, said the draft text on Ireland committed Britain to "full alignment" on rules.
Donald Tusk, the EU summit chair, tweeted ebulliently after speaking to Varadkar that there was good progress on the Irish issue to unblock UK-EU trade talks: "Tell me why I like Mondays!" the former Polish premier wrote. "Getting closer to sufficient progress at December (EU summit)."
This is code for Britain complying with a set of conditions the EU wants met on divorce terms before leaders will agree to launch talks on a future trade deal at a summit next week.
Tusk, who had cancelled a trip to Jerusalem and Ramallah because of apparent progress on Brexit, met May after her talks with Juncker. Officials had earlier said he was preparing to call round EU leaders to get agreement on trade negotiations. National envoys handling Brexit were also summoned urgently to a meeting in Brussels later on Monday.
London has broadly agreed to many of the EU's divorce terms, including paying out something like 50 billion euros. But the issues of the rights of expatriate citizens and the UK-EU border on the island of Ireland have defied a deal.
Elmar Brok, another member of the European Parliament who met Juncker and his Brexit negotiator, Michel Barnier, before they met May, said "just a few words" separated the sides. (Additional reporting by Andrew MacAskill, Estelle Shirbon and Kate Holton in London, Alastair Macdonald, Lily Cusack and Jan Strupczewski in Brussels and Conor Humphries in Dublin, writing by Alastair Macdonald; Editing by Larry King, Andrew Heavens and Peter Graff)