This would keep Northern Ireland under EU economic oversight if London and Brussels can not agree a trade pact to keep UK-EU borders open after a transition period ends in 2020 - an idea that May and a small party in the province that props up her minority government oppose. The Remain side edged the vote in Northern Ireland with 55.77% of the vote. The U.K., she said, would be waiting for the E.U.to come back with reasonable proposals. "The U.K. expects the same ... at this late stage in the negotiations, it is not acceptable to simply reject the other side's proposals".
Cabinet colleagues reiterated her message on Friday.
"Neither side should demand the unacceptable of the other, we can not accept anything that threatens the integrity of our union as they can not accept anything that threatens the integrity of theirs".
The Sun went much further: "EU Dirty Rats - Euro mobsters ambush May", it said alongside a mock-up picture of French President Emmanuel Macron and European Council President Donald Tusk dressed as American gangsters with guns.
Mrs May added: "It is something I will never agree to - indeed, in my judgement it is something no British Prime Minister would ever agree to".
In a possible sign that she was smarting over Mr Tusk's mocking Instagram photo, showing him offering her a cake with the caption "Sorry, no cherries", Mrs May made a point of stressing that she had always shown respect towards her European Union counterparts.
"If all conventional roads lead to a hard no-deal Brexit, the notion of Parliament exerting control and forcing another referendum on us would begin to look not wholly fanciful", Peston wrote in the Spectator.
Many who took part in the meetings were surprised by a near unanimous verdict in Britain that the prime minister had taken an orchestrated drubbing in the city of Mozart and the "Sound of Music"; but if European Union leaders did sound a touch sharp and careful to sound in tune with each other, that reflected a genuine reaction to comments and tactics from May and her ministers this week.
"The trouble with the position the European Union took yesterday is it pushes people like me further into the camp of those who say "look, we told you so, there is no compromise or flexibility to be found on the European Union side".
In a speech in Munich, the former Brexit secretary launched a scathing attack on the UK's "continental friends", singling out German politicians for wanting to see Britain fail after it leaves the EU.
Tusk said May has just four weeks to amend her plan - or perhaps face the dreaded prospect of leaving Europe with no deal - a so-called doomsday scenario. You are our friends, our neighbours, our colleagues.
"Second, I want to reassure the people of Northern Ireland that in the event of no deal we will do everything in our power to prevent a return to a hard border". To deny its legitimacy, or frustrate its result, threatens public trust our democracy.
Turning to the consequences of no-deal between the United Kingdom and the EU, Davis warned that as well as the auto industry being hit, the German dairy, French drinks and Irish beef trades would also suffer. She called for "serious engagement" to solve the problems.