Two leading Brexiteer cabinet ministers, David Davis and Liam Fox, have told her that plans to combine a customs partnership and technological checks at the UK’s borders cannot be delivered, with Mr Fox said to be considering his position.
Meanwhile, the Scottish Government has repeated its call for the Prime Minister to keep the UK in the single market and customs union, with the SNP’s Brexit minister Michael Russell telling Mrs May to “grow up” and “wake up to reality”.
Mr Russell held talks with UK counterparts in London, and said afterwards that the Scottish Government had no confidence that a draft white paper set to be approved by Mrs May’s Cabinet on Friday could even be delivered.
Instead, the government in Edinburgh has produced its own, rival white paper repeating its demands to stay in the single market and customs union, and maintain free movement between the UK and the EU.
“What we read of the solutions they are putting forward are neither practical or workable.”
He added: “This is a comprehensive document…. it says how we should be dealing with issues and we are trying to say to the UK Government, please grow up, listen to what is real.
“I was in Strasbourg earlier this week. I spoke to members of the European Parliament and members of the [EU] Commission. Everybody uses the same word - reality. The UK Government has to wake up to reality.”
Scottish Secretary David Mundell pledged to put forward the Scottish Government's proposals at Chequers, even though he said "on a fundamental level, we don’t agree because the Scottish Government doesn’t want to leave the EU."
"Leaving the EU is about leaving the single market because it’s not possible to be in the single market and not be in the EU," he said. "I’ll be arguing strongly that our businesses get the best possible access to those markets."
Mr Mundell added that his "fundamental requirement" was the "integrity of the united kingdom as we leave the EU" and said he was "not going to agree or let happen” any backstop plan for the Irish border that creates trade barriers in the Irish sea between Norther Ireland and Britain.
The Scottish Secretary appealed for unity among his cabinet colleagues, warning that "we have to have a united position in this country as to what we want to achieve, because that’s the most likely way of getting what we want."
However, on the eve of the day-long talks at the Prime Minister’s Buckinghamshire retreat, ministers voiced skepticism about the so-called "third way" to solve problem of post-Brexit customs and preventing a hard border with Northern Ireland.
A proposed "facilitated customs arrangement" would see technology used to determine where goods arriving into the UK will ultimately end up.
This would allow the correct tariff to be paid - either at the UK or EU rate.
Officials in Brussels are concerned about the reliance on untested technology, with reports they have described the plan as "an open invitation to smugglers".
Mrs May flew to Berlin on Thursday for talks on Brexit with Chancellor Angela Merkel, in a bid to head of a rejection from Brussels.
Amid growing speculation that Mrs May could be forced into a climbdown over her Brexit red-lines, a group of more than 40 Eurosceptic Tories met Chief Whip Julian Smith on Wednesday to air their concerns about the prospect of a soft Brexit which would restrict the UK's freedom to diverge from EU rules in future.
A meeting of the European Research Group of Eurosceptic Tory backbenchers this week was reported to have been the "angriest… even stretching back to Maastricht" according to one MP in attendance.