May, her Brexit minister David Davis and the prime minister's Brexit adviser Olly Robbins, will meet with EU Commission Jean-Claude Juncker, EU negotiator Michel Barnier and Junkcer's chief of staff, Martin Selmayr, for lunch.
After a day that see-sawed between hope and disappointment, the leaders failed to make what the European Union considers "sufficient progress" on three issues: Britain's exit bill, the rights of citizens affected by Brexit and the status of the now invisible Irish border.
In a joint press conference following the meeting, both leaders tried to put a positive gloss on the outcome, saying they were within touching distance of a deal.
London has broadly agreed to numerous EU's divorce terms, including paying out something like 50 billion euros.
The Irish Times are reporting that the "agreement is based on maintaining "regulatory alignment" between Northern Ireland and the Republic after Brexit".
He added: "The indications we have is that we are in a much better place now than we have been in the negotiations to date".
After Britain leaves the bloc, the now invisible 310-mile (500-kilometer) frontier will be the U.K.'s only land border with an European Union country.
The UK however said it would not create a border between Northern Ireland and the rest of Britain.
However, unionists tend to view with scepticism any proposal for harmonisation of rules on either side of the border.
Dublin has claimed the United Kingdom government has not yet tabled a serious or credible plan to avoiding a hard border between the Republic of Ireland and Northern Ireland after Brexit, if the United Kingdom quits the EU's single market and customs union.
Irish and European Union officials indicated that agreement was close.
The DUP has only 10 seats out of 650 in Britain's House of Commons, but without their support May's government would fall.
"We will not accept any form of regulatory divergence which separates Northern Ireland economically or politically from the rest of the United Kingdom", she said. "The economic and constitutional integrity of the United Kingdom must not be compromised in any way". "If no clear commitment is made, the EPP group will not be ready to assess the progress made as sufficient to enter a 2nd phase of negotiations", Manfred Weber, German MEP and leader of the largest group, the European People's Party in the EP, tweeted.
"And the problem between the Republic and Northern Ireland is a outcome of the British decision not only to leave the EU but also the single market and the customs union".