The agreement would avoid the need for an Irish backstop - which has left Britain and Brussels deadlocked - and would stop Northern Ireland being treated differently to the rest of the UK.
According to the Telegraph, May has included Attorney General Geoffrey Cox, the government's chief legal adviser, in her special Brexit Cabinet after ministers said they wouldn't sign off on a deal without his advice.
In a major intervention on the controversial backstop, amid reports that the government plans to keep Northern Ireland in aspects of the European Union trade structures, Mr Davis said it was "pretty clear there is genuine and significant concern regarding the implications of any fresh backstop text".
The EU has proposed a backstop that would mean Northern Ireland staying in the EU customs union, large parts of the single market and the EU VAT system.
Speaking at a meeting of the British-Irish Intergovernmental Conference in Dublin, Mr Coveney said he believed "it is possible to get a deal in November", adding: "This border issue is complicated to resolve. but I think we're very close to resolving it".
May is also understood to be close to a political deal on a future economic partnership (FEP) with Brussels that would give Britain the green light to pursue a free trade deal similar to Canada.
A Downing Street spokesman called the newspaper report "speculation".
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May's deal, based on the Chequers plan which prompted Johnson and former Brexit Secretary David Davis to resign from her Cabinet, would see Britain agree to a "common rulebook" with the EU on goods and agri-products - identical to the EU rulebook, and interpreted by EU judges - and remain inside the bloc's Customs Union, as a so-called "backstop" solution to the vexed question of keeping the Irish border open.
The PM has secured "private concessions" from Brussels that the whole of the United Kingdom will be allowed to remain in a customs union with the EU after Brexit occurs in March, according to The Sunday Times' political editor Tim Shipman.
It was claimed that the European Union had last week agreed a major concession to unlock a Brexit deal, by accepting that checks on goods could take place in factories and shops rather than at the border.
It said Mr Raab made the proposal to Mr Coveney in London last week.
It is speculated that May is hoping for enough progress in Brexit talks this week to secure a summit later this month in which the final details of a deal will be negotiated. "I certainly hope we are".
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.