The febrile atmosphere in Westminster comes as negotiators in Brussels attempt to reach a deal on the withdrawal agreement.
But the result, European Union officials and diplomats said, is that there are no plans for further talks before leaders of the other 27 European Union member states meet over dinner in Brussels on Wednesday to hear Barnier brief them on the state of play.
Brussels has suggested that in the event of no deal, Northern Ireland should stay in the EU customs union - something the United Kingdom has repeatedly rejected as it would create a new border between Northern Ireland and the United Kingdom across the Irish sea.
But after talks which lasted a little over an hour, it was clear that obstacles remained.
Government tensions are running high over the precise wording of a "backstop" deal on customs to prevent the imposition of a "hard border" on the island of Ireland.
The issue of the border between Northern Ireland and Ireland is one of the last remaining obstacles to achieving a divorce deal with Brussels, with wrangling continuing over the nature of a "backstop" to keep the frontier open if a wider UK-EU trade arrangement can not resolve it.
A British government spokesman said progress had been made in some key areas.
But despite suggestions that negotiators had come to a tentative agreement, news broke that key issues including the Irish border question remained unresolved.
With continued speculation that Leave-supporting Cabinet ministers could resign if Mrs May presses ahead with the plan, David Davis said: "This is one of the most fundamental decisions that government has taken in modern times".
Arlene Foster, the DUP leader, said on Thursday afternoon that the prime minister could not "in good conscience" recommend such a deal and vowed to oppose it.
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In case we don't get a future trade deal, the European Union wants what has become known as the "backstop" as an insurance policy - something that was initially agreed by the UK Government and the European Union last December.
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.
"There are different ways to ensure that something is time limited. This week the authority of our constitution is on the line", writes Mr Davis, who stepped down from the UK Cabinet in July due to his opposition to Ms May's Brexit strategy agreed by the rest of the Cabinet at her country residence of Chequers.
The Sunday Times said at least nine ministers want Mrs May to change course when the Cabinet meets on Tuesday.
It has been reported that four more letters demanding a vote of no confidence in the PM were submitted last week after three during the Tory conference, making a total of 44, just four short of the number needed to trigger a vote.
Critics of the Prime Minister believe that the threshold could be passed by Wednesday depending on the nature of her approach to the Brexit talks.
But she added: "Getting May out and him becoming an interim leader may be the only way to deliver Brexit and FTA (a free-trade agreement)".
Ms Dorries said that Mr Davis had always focused on changing the policy, rather than the Prime Minister.
Someone who has looked into this so you don't have to is Lars Karlsson, the former Director of the World Customs Organisation, who was asked to investigate solutions by the European Parliament - hardly a pro-Brexit organisation!