Reports at the weekend suggested that the EU is now ready to contemplate concessions which would keep all of the United Kingdom in a temporary customs union following the end of a transition period due to end on December 31 2020.
But that has not stopped a flurry of concern from backbench Tory Brexiteers who are concerned the cabinet may be about to be "bounced" into agreeing to terms that they say could bind the United Kingdom into an indefinite customs "limbo" with the EU.
After meeting Friday with Irish Deputy Prime Minister Simon Coveney in Dublin, British Prime Minister Theresa May's deputy David Lidington said negotiators were "very close" to an agreement.
Downing Street dismissed the report as "speculation" and repeated its claim that the deal was "95 per cent" done - but did not deny the substance outright.
Theresa May has been warned by the Irish prime minister Leo Varadkar that his government will never accept a Brexit deal that gives the United Kingdom unilateral power to end a Northern Irish "backstop".
The plan sees the whole of Britain remaining in the EU customs union - not just Northern Ireland.
According to the Sunday Times, the agreement would include an "exit clause" aimed at convincing Brexiteers that remaining in the customs union would be temporary.
Under the concessions, the European Union is said to have accepted that regulatory checks on goods be allowed to take place "in market" - effectively meaning that checks can be done in places like factories, rather than at the border between the two countries.
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A spokesperson said: 'The prime minister has been clear that we are making good progress on the future relationship and 95 percent of the withdrawal agreement is now settled and negotiations are ongoing'.
The newspaper also said May is on course to secure an agreement on a "future economic partnership" that would let Britain keep open the prospect of a free trade accord similar to the one Canada has with the EU.
Senior sources told the paper that May has secured concessions from Brussels, with the EU agreeing to write an "all-U.K." customs union into the divorce deal.
Meanwhile, the report predicts that the PM will sell the deal to hardliners within her party by insisting that they will be to blame should the United Kingdom crash out of the European Union without a deal in four months' time.
Even if a deal is done in Brussels in the coming days, May will have to sell it in London - first to her own Cabinet, and then to Parliament.
"This is the backstop", he added, saying the United Kingdom had agreed that it would apply "unless and until" a close future relationship eliminated any need for Border infrastructure and checks. "That is why we are increasingly positive on the expectation of reaching a deal", he added.
"While we too hope the Northern Ireland backstop will never be required to be used, it will be required to be written down in legal text".