Downing Street has produced a third possible model for the UK’s post-Brexit customs arrangements, which will be among issues discussed by the cabinet at a crunch meeting at Chequers on Friday.

No details have been released about the possible customs plan, but it is distinct from the two options presented so far – the maximum facilitation option, and the customs partnership – the Guardian understands.

Both the initial ideas have run into trouble. The customs partnership, under which the UK would collect EU import tariffs, has been dismissed as unworkable by Brussels and is intensely disliked by Brexiters, while “max fac” relies on complex and uncertain technological solutions.

No 10 sources suggested that the customs partnership, which was previously May’s preferred option, was “dead”.

It raises the likelihood of the new third model retaining key features of max fac, but to be introduced further down the line.

One remain-supporting cabinet minister said that they did not believe max fac in its current form would ever happen. “The reality is that this can will be kicked further down the road, probably to a post-May administration.”

A third option will be seen as an attempt by Theresa May to break at least part of the deadlock as her ministers gather for a day-long retreat at the prime minister’s country home in an attempt to thrash out a vision for the UK’s future relationship with the EU.

The decisions reached at the meeting will form the basis for an imminent and long-awaited white paper.

With hopes fading of a bespoke wider agreement, senior government sources say ministers are broadly being asked to choose between an intimate, Norway-style partnership and a much looser, Canada-style trade deal.

Leading cabinet Brexiters such as David Davis and Boris Johnson are vehemently resisting the idea of a Norway-type deal, which would leave the UK subject to some EU rules.

However, allies of the foreign secretary and the Brexit secretary denied they were poised to resign if they lost Friday’s debate. A friend of Johnson’s said: “He’s not going anywhere: he’s always been of the view that he has more influence in the room.”

Davis, who has long-favoured what he calls a “Canada plus plus plus” approach – echoing Ottawa’s trade deal with the EU but covering services – has held a series of meetings with May in recent days.

On Monday, the leading backbench Brexiter Jacob Rees-Mogg warned that for May to choose a softer option would prompt a rebellion among her MPs. The move would bring a party crisis on a scale of the repeal of the corn laws, he said.

“Theresa May must stand firm for what she herself has promised,” Rees-Mogg wrote in an article for the Telegraph. “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.”

The housing minister, James Brokenshire, told the BBC’s Andrew Marr Show on Sunday that the government was determined to reach a clear resolution this week – and the outcome would be published in the white paper.

“I think there’s no doubt that there are strong views on either side, and that’s what I would expect as we lead into the discussions on Friday,” he said. “But equally, I remain confident that we will come out from that meeting with that clear direction set out, and the white paper that will follow.”

May will report back to MPs on Monday about last week’s European council meeting, at which leaders made a series of comments about indecision and divisions in London.

The 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.”

Cabinet tensions have mounted in recent days, with collective responsibility all but dissolving, amid widespread speculation that several leading members are burnishing their credentials for a potential leadership bid should May be forced out.