But the DUP pulled the plug as it raised fears that the United Kingdom was making too many concessions when it came to the Irish border.
Meanwhile, Sinn Fein's Brexit spokesman David Cullinane told Sky News it is "crucial" Northern Ireland remains in the customs union and single market as "the only way to avoid the hardening of the border". Both options risk alienating large elements of the Conservative Party and hard Brexiteers.
The DUP leader, Arlene Foster, had made public her anger at the suggestion, and it is understood that that message was repeated in a call with the prime minister on Monday afternoon.
"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. But then Northern Ireland's Democratic Unionist Party, which props up May's minority government, announced it wouldn't support any deal that made special rules for Northern Ireland.
The failure to seal a deal threatens to delay the progression of the Brexit negotiations to the second phase covering trade and the UK's future relationship with the European Union (EU).
An EU official, who will be consulted by Mr Barnier and Mr Juncker ahead of Monday's lunch with the Prime Minister, added "it was more likely than unlikely" that an agreement on withdrawal issues will be reached.
Negotiations could possibly go on until Wednesday when Barnier will brief European Union commissioners about the progress and make a formal recommendation on whether "sufficient progress" has been achieved for the next phase of talks to be unlocked.
"And it is clear, crucially, that we want to move forward together".
"Crucially it is clear that we want to move forward together, but on a couple of issues, differences do remain, which require further negotiation and consultation", said May before heading to meet President of the European Council Donald Tusk nearly one-hour late than scheduled.
The UK government has repeatedly insisted many of these questions can not be answered until the next stage of Brexit talks - on a transition period and a future EU-UK trading and customs relationship - is allowed to begin.
"I hope we are in a place this evening where Irish people north and south will get reassurance from the wording that is very close to being finalised now".
"We now have a common understanding on most related issues with just two or three open for discussion".