Downing Street has produced a third model for handling customs after the UK leaves the EU, the BBC understands.
Details of the new plan have not been revealed publicly but senior ministers will discuss it at Chequers, the prime minister’s country retreat, on Friday.
Ministers have been involved in heated discussions recently as they tried to choose between two earlier models.
Tory backbencher Jacob Rees-Mogg says the PM risks a revolt if the type of Brexit she promised is not delivered.
The PM is set to be pressed to give more detail of the plan when she faces MPs in the House of Commons from about 17:00 BST.
Writing in the Daily Telegraph, Mr Rees-Mogg said he and other members of the 60-strong group of Eurosceptic Tory MPs he leads, known as the European Research Group, would reject a deal that did not amount to a clean break with the EU.
But amid signs of widening Conservative divisions, Foreign Office minister Alan Duncan suggested the “insolent” MP should “pipe down”.
Theresa May hopes to resolve cabinet splits on the shape of Brexit at this week’s meeting.
Her official spokesman refused to be drawn on there being a third customs model saying: “There is going to be a lot of speculation between now and Chequers.
“Some of it might even be true, but I’m not going to engage in advance of the away-day taking place.”
The prime minister has said that after the cabinet gathering the UK will publish a White Paper setting out “in more detail what strong partnership the United Kingdom wants to see with the European Union in the future”.
Reality Check: The government’s customs options
Ministers ‘will agree on Brexit stance’
Kuenssberg: All eyes on Chequers
Brexit: All you need to know
It follows last week’s summit in Brussels where European Council president Donald Tusk issued a “last call” for the UK to agree its position on Brexit, saying the “most difficult” issues were unresolved and “quick progress” was needed if agreement was to be reached by the next meeting in October.
BBC political correspondent Chris Mason says Downing Street hopes it has now found its way out of a bind on customs, the issue central to the practicalities of the UK’s future trading relationship with the EU, and a significant part of finding a solution to maintaining an open border with the Republic of Ireland.
The UK is due to leave the EU on 29 March 2019, and negotiations are taking place on what the future relationship between the UK and the EU will look like.