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.
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.
The Prime Minister had reportedly persuaded European Union bosses to drop their insistence on splitting up the United Kingdom and separating Northern Ireland from the rest of Britain's Home Nations with customs checks in a so-called "customs compromise".
The backstop is an arrangement that applies if the Irish border can not be kept frictionless during a wider deal.
Last night Downing Street sought to play down expectation that a customs union deal was close to being agreed.
The Labour Party has all but ruled out supporting any deal May reaches with Brussels, leaving her reliant on her slender parliamentary majority, which comprises her own divided MPs and a small coalition partner, the Democratic Unionist Party.
The deal keeps open the possibility of Britain securing a future free trade agreement similar to the one signed between Canada and the European Union in 2016.
Some at the highest levels of government fear that, unless progress is agreed by Tuesday when the May sees her senior ministers and parliament breaks for recess, the cabinet may not have a direct input before a summit announcement is made.
"The Democratic Unionist Party is rightly anxious about the future of the Union and many Conservatives are too".
"The prime minister has been clear that we are making good progress on the future relationship and 95% of the withdrawal agreement is now settled and negotiations are ongoing", the spokesman also said, as quoted by the Sky News.
The Telegraph said the proposal was "apparently contradicted" by Cabinet Office Minister David Lidington on a visit to Dublin.
"As soon as MPs understand what is really at stake, I have no doubt that they will throw this deal out".
But some of Britain's biggest corporate names have had enough of the government's handling of the negotiations.
'They will be bad for business and bad for working people, ' the letter said.
Reports of a secret Brexit deal come just as more than 70 business leaders signed a letter calling for a second referendum after warning that a bad deal could threaten the United Kingdom jobs industry.
'Given that neither was on the ballot in 2016, we believe the ultimate choice should be handed back to the public with a People's Vote'.