Donald Tusk, the EU summit chair, tweeted ebulliently after speaking to Irish Prime Minister Leo Varadkar that there was progress on the Irish issue to unblock UK-EU trade talks: "Tell me why I like Mondays!" the former Polish premier wrote.
London has broadly agreed to numerous EU's divorce terms, including paying out something like 50 billion euros.
"This is not a failure", said Juncker.
The EU has had "enough time now to decide whether or not they are going to discuss trade with us, they need to get on with it and if they don't get on with it, the closer we get to walking away with no deal", she said.
Juncker described May as a "tough negotiator", and insisted that talks had not failed.
In addition, the committed Brexiteers among the DUP argue that if, over time, Northern Ireland diverges from the trading regulations applied elsewhere in the United Kingdom, it could create a barrier to trade within the United Kingdom and prevent Belfast benefiting from any deals London might negotiate in the future.
Tusk, canceled a trip to Jerusalem and Ramallah and hastily scheduled his own meeting with May for after lunch.
But, asked if he expected an agreed text of written British assurances on Monday, Mr Coveney told RTE Radio: "Let's not run before we can walk here".
Britain is set to leave the European Union on March 29, 2019 yet the progress on its exit and the terms of new relations have been painfully slow. The pro-British Unionist party opposes any special status that could take Northern Ireland further from Britain and closer to the Republic of Ireland.
Ms Foster had issued a statement warning it would not accept "any form of regulatory divergence which separates Northern Ireland economically or politically from the rest of the UK".
Juncker and Barnier will meet the European Parliament's Brexit team at 11 a.m.to brief them on progress.
But the two sides said they were within striking distance of consensus, setting up a hectic negotiating rush ahead of an European Union summit next week.
May portrays Monday's meeting as part of preparations for an European Union summit on December 15 - though the European Union says no negotiations will be conducted at the summit itself, so Monday is the last chance for her to make offers.
"And the problem between the Republic and Northern Ireland is a effect of the British decision not only to leave the EU but also the single market and the customs union".
With the clock ticking down to the March 2019 exit date, May is under pressure to start talks on its future trade ties by the end of the year to remove the cloud of uncertainty for companies that do business in the EU.