Sufficient progress has been made in Brexit talks, European Commission President Jean-Claude Juncker has said; paving the way for talks on the future UK-EU relationship.
The pro-British DUP party which props up May's government refused Monday's draft deal over a phrase suggesting there would be "regulatory alignment" with the European Union after Brexit - effectively putting Northern Ireland under European Union law.
A new round of talks between Brussels and London is relatively Brexit is expected today.
It comes after Theresa May and David Davis headed to Brussels to meet with Mr Juncker and the EU's chief Brexit negotiator, Michel Barnier.
Conversations continued with the DUP, which scuppered a deal on the Irish border on Monday after balking at "regulatory alignment" planned between Northern Ireland and the Republic. "Talks are continuing throughout the night".
Yesterday the Commission's spokesman Margaritis Schinas indicated that window could be pushed - albeit slightly - as there was no "white smoke" on the deal yet.
May's office said she and Varadkar are "working hard to find a specific solution to the unique circumstances in Northern Ireland", which have triggered a crisis in Britain's divorce negotiations.
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"We believe there is still more work to be done to improve the paper".
"I just don't see either side saying it's all fallen apart & back to the drawing board", says Neil Jones at Mizuho Bank in London, "the bottom line is Brexit breakthrough is to send the Pound higher".
In that scenario, the leaders would once again rule that insufficient progress had been made on the opening issues of citizens' rights, the financial settlement and the Irish border for talks on trade and a transition period to start.
Mrs May has said she wants a deep and special partnership and a better deal than the free-trade agreement that Canada secured from the EU.
The Democratic Unionist Party (DUP) insists that Northern Ireland, as part of the United Kingdom, must leave the European Union in the same way as the rest of the United Kingdom.
The DUP objected to plans for "regulatory alignment" between Northern Ireland and the Republic to maintain a soft border between the two, arguing it would amount to the drawing of a new frontier with the United Kingdom mainland in the Irish Sea.
She must also convince her divided Conservative Party that the deal she makes is acceptable.