At the same time, it reported that the prime minister was on course to agree a future economic partnership that would leave open the possibility of Canada-style free trade deal sought by Brexiteers.
A Downing Street spokesman called the newspaper report "speculation".
Theresa May appears to have accepted that an open border with the Irish Republic is absolutely vital - despite its regular exploitation by organised criminals and illegal migrants - but can not leave Northern Ireland in the Customs Union by itself, as this would lead to customs checks between Britain's Home Nations.
The Prime Minister is said to be preparing to sell the plan to Brexiteers in the Cabinet by insisting the customs arrangement will not be permanent, and that the alternative is a potentially disastrous "no-deal" exit from the bloc in March.
Plan removes the need for a so-called "backstop" on the Northern Irish border issue.
The Sunday Times claims the Prime Minister is optimistic that her secret plan will convince some remainer Conservatives and swithering Labour MPs to back the deal.
The EU has reportedly accepted that customs checks can take place at shops and factories, rather than at the border - which, if true, would remove the problem of where to put the customs border post-Brexit.
"Negotiations are still very firmly continuing, and therefore we are 95% of the way there in relation to the Withdrawal Agreement".
The Sunday Times reports that the prime minister is close to a political deal on the United Kingdom and EU's so-called future economic partnership, which the newspaper says will "allow Britain to keep open the prospect of a free trade deal resembling that enjoyed by Canada".
It said Mr Raab made the proposal to Mr Coveney in London last week.
"This has been committed to by the United Kingdom in order to have a Withdrawal Agreement".
'I am assuming the Government will stick to its manifesto commitment to leaving the customs union when we leave the EU'.
It came as more than 70 business leaders backed a referendum on the terms of the Brexit deal, warning that the United Kingdom faces "either a blindfold or a destructive hard Brexit" that would be bad for both firms and jobs.
In a letter to Mrs May they say Parliament should not be bound by the 2016 vote any more than it should be by the 1975 referendum that took Britain into the European Union, especially when there are question marks over its validity.
Waterstones chief executive James Daunt, ex-Sainsbury's chief executive Justin King, Lastminute.com founder Baroness Lane-Fox and Innocent Drinks co-founder Richard Reed are among over 70 signatories of a letter calling for a second Brexit referendum.