With just five months until Britain is due to leave the EU, May has yet to nail down a divorce deal, with the insurance arrangement to keep open the border between British-ruled Northern Ireland and EU member state Ireland still the outstanding issue.
However, sources said there was immediate disagreement over whether the proposal would give the United Kingdom the power to unilaterally call time on the backstop, with both the home secretary, Sajid Javid, and the foreign secretary, Jeremy Hunt, arguing that it must. The agreement would "not be done at any cost", her official spokesman said.
Hardliner Michel Barnier was characteristically belligerent, insisting that the "backstop means backstop" and there is "still a real point of divergence" on the Northern Ireland issue.
May said she would not agree a withdrawal "at any cost", and that it would depend on an "acceptable" framework for future relations in areas like trade and security, expected to be covered in a separate political declaration.
He said: "Don't be under any illusion, there remains a significant amount of work to do".
Ministerial sources told HuffPost UK the cabinet could approve the long-awaited Withdrawal Agreement from the European Union as early as Thursday or Friday this week, followed by a special Brussels summit later this month and a Commons vote in December.
"We're willing to consider improvements to the backstop but we need to reach an agreement for this backstop and this backstop must be a genuine backstop".
The Prime Minister's spokesperson said ministers needed more time to consider possible mechanisms that will prevent the United Kingdom from being tied to the European Union through a backstop arrangement to prevent a hard border in Northern Ireland.
In response, Sir Jeffrey said Dublin's stance was making a no-deal Brexit likely. The UK can not negotiate its own trade agreement while it is signed up the the customs union.
The notes say ministers would seek to claim "measured success", a deal that is "good for everyone", rather than claiming a victory with champagne corks popping.
Prime Minister Theresa May called a meeting Tuesday in what pundits say was a last ditch attempt at persuading colleagues to back a plan that could ultimately see all of the United Kingdom remaining inside the EU customs union, with all that would entail, in the event a Brexit-trade deal cannot eventually be reached.
Donaldson said that he could not understand why the Irish government "seems so intent on this course".
He tweeted: "Such an outcome will have serious consequences for economy of Irish Republic".
Looks like we're heading for no deal.
This shows the increasing fears from European Union regulators over the consequences of a no-deal Brexit and the possible significant impacts on the operation of clearing houses, which have become central to a study financial system since the start of the recession in 2008.