Britain's Brexit pointman Dominic Raab made an unannounced trip to the European Union capital for talks with his opposite number Michel Barnier, briefly raising hopes that a deal might be in the offing.
"And delivering this is vital for our security too because economic relations sit at the heart of all our partnerships including the vital security alliances we have with our friends in Europe".
Health Secretary Matt Hancock sought to play down the potential for Cabinet resignations but was unable to say whether a fixed deadline for any customs arrangement would be written into a deal with Brussels.
Less than six months before Britain leaves the bloc and days before May heads to Brussels for a summit on Wednesday when both sides hope to make progress, the Brexit talks were paused on Sunday after the two sides failed to agree on how to deal with the United Kingdom's only land border with the EU.
Mrs Foster said the EU's backstop suggestion whereby Northern Ireland would continue to follow the bloc's rules if no trade deal is struck would create barriers with Great Britain. We will be outside the European Union but run by the European Union, in the sense that from next year we will, of course, have no one in the Commission directorate of external trade, no influence over tariffs, and no ability to decide what trade and commercial policies will be pursued IN OUR OWN COUNTRY.
Sunday's talks ended without a breakthrough on the crucial issue of trade to and from Northern Ireland, which has emerged as a possible deal-breaker and even a threat to May's leadership.
Brexit negotiations with the European Union have accelerated and become more positive over the past week, though significant hurdles remain, finance minister Philip Hammond said.
Up until now, the Prime Minister has repeatedly warned that a failure to back her Brexit strategy would result in a disorderly withdrawal from the bloc.
The DUP leader said proposals for a "backstop" arrangement - effectively keeping Northern Ireland in the single market and custom union to avoid a hard border on the island of Ireland - would lead to its "permanent annexation" from the UK.
"We can not let this disagreement derail the prospects of a good deal and leave us with a no deal outcome that no-one wants", Mrs May told parliament.
The talks came ahead of a private meeting between DUP leader Arlene Foster and Taoiseach Leo Varadkar in Dublin today.
With the negotiations coming to a head, the central focus of the discussions on Thursday is thought to have been the issue of the Northern Ireland "backstop" meant to ensure there is no return of a "hard border" with the Republic.
However, several senior Cabinet ministers threatened to walk out over the deal with Conservative MPs plotting to install former Brexit secretary David Davis as prime minister.
The need to resolve the backstop issue is a political headache for the Prime Minister, who depends on the votes of the 10 DUP MPs to prop up her administration in Westminster.
"Nobody wants to ever trigger the backstop, but it needs to be there as an insurance mechanism to calm nerves that we're not going to see physical border infrastructure re-emerging", Coveney said.
Instead, he said, Britain should pursue an enhanced version of the Canada free trade deal with the EU. But EU officials say that may be the only way to avoid a hard border between Northern Ireland and Ireland.
Johnson, who regularly uses his newspaper column in the Daily Telegraph to excoriate May's Brexit plan, said the EU's border backstop amounted to "a choice between the breakup of this country or the subjugation of this country, between separation or submission". "Whether or not we do this week, who knows?"
Britain's global trade, environment and Brexit ministers told May at a meeting on Thursday that they fear the whole of Britain could remain in the EU customs union for an open-ended period, the BBC said.