The Prime Minister is due to discuss the customs model with ministers this week, after two earlier proposals sparked Cabinet divisions.
Downing Street officials have drawn up the third model ahead of a crunch Cabinet away-day at Mrs May’s Chequers retreat on Friday.
No details about the proposal have been revealed, but Mrs May will hope it will prove more popular than the two other options.
She favours a ‘customs partnership’ with Brussels, which would see Britain collecting EU import tariffs.
But this has proved hugely unpopular with the Cabinet’s Brexiteers including Foreign Secretary Boris Johnson.
Mr Johnson prefers the ‘max fac’ option, which aims to use technology to remove the need for physical customs checks.
Cabinet tensions have soared in recent days, with in-fighting over the thorny issue of Brexit showing no signs of abating.
And influential backbencher Jacob Rees-Mogg today warned Mrs May could be forced out of office if she fails to deliver Brexit.
The chairman of the pro-Brexit European Research Group said: “Theresa May must stand firm for what she herself has promised.
“One former Tory leader, Sir Robert Peel, decided to break his manifesto pledge and passed legislation with the majority of his party voting the other way.
“This left the Conservatives out of office for 28 years. At least he did so for a policy that works.
“At Chequers, Mrs May must stick to her righteous cause and deliver what she has said she would. She must use her undoubted grace to persevere.”
At last week’s European Council summit, EU leaders made a series of comments about indecision and divisions in London.
European Commission president Jean-Claude Juncker, said: “I don’t have to lecture Theresa May, but I would like our British friends to make clear their position.
“We cannot go on to live with a split Cabinet. They have to say what they want and we will respond to that.”
European Council leader Donald Tusk added: “There is a great deal of work ahead, and the most difficult tasks are still unresolved.
“If we want to reach a deal in October, we need quick progress. This is the last call to lay the cards on the table.”