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.
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.
The Sunday Times has been told that talks over a final deal are far more advanced than previously indicated, with a 50-page document likely to be published, rather than the five-page plan many were expecting.
Source close to EU negotiator Michel Barnier revealed that a “major concession” had been made on the Irish border during a meeting in London last week that will see regulatory checks on goods carried out at factories and shops rather than being conducted at the border.
• READ MORE - Vince Cable talks Brexit with drink giant Diageo
A special summit is understood to have been pencilled in for later this month to agree the final details of a deal.
May is set to discuss the proposals with her Cabinet on Tuesday, and is likely to attempt to win backing for the plan by warning Brexiteer ministers that they will be personally responsible for a “no-deal” Brexit if they fail to support it and telling remainers that, while she fought for the closest possible alignment until the end, negotiations mean Britain will not be a member of the customs union indefinitely.
The reports suggest that the Prime Minister has effectively abandoned her Chequers plan, and is now open to negotiating a different “landing zone” for the final deal.