Supporters of a "Norway-plus" plan for close UK engagement with the EU after Brexit have been boosted by support from both Labour and the Scottish National Party.
The scheme - known as Common Market 2.0 - is one of nine alternative plans tabled for a series of indicative votes on Monday designed to establish what kind of Brexit has a chance of winning a majority among MPs.
Conservative MPs have been given a free vote, but Cabinet ministers will be told to abstain.
The Norway-plus proposal, put forward by Conservative Nick Boles, was defeated by 94 in the first round of indicative votes last week, when Labour MPs were "encouraged" to support it but the SNP abstained.
It proposes UK membership of the European Free Trade Association (Efta) and European Economic Area (EEA), allowing continued participation in the single market and a "comprehensive customs arrangement" with the EU after Brexit.
If all 35 SNP MPs fall in behind the plan on Monday evening, it could win a majority unless 7-8 or more Labour MPs rebel against their party's whip.
Meanwhile, it was confirmed that senior ministers will meet in Cabinet for five hours on Tuesday - the first three hours in a "political" session in the absence of civil servants - to discuss the way ahead, amid speculation about possible resignations, a general election or change of Tory leader.
And reports from Brussels suggested that the EU is seeking more than 10 billion euros from the UK to cover liabilities for the rest of this year, even if Britain leaves without a deal on April 12.
Irish broadcaster RTE quoted an unnamed "senior EU source" as saying Brussels hopes to "wrap up" agreement on the payment by the end of the week.
Labour MPs will be told to support the party's own alternative Brexit plan - tabled as an addition to the order paper on Monday morning - as well as the Common Market 2.0 scheme, Tory grandee Kenneth Clarke's plan for a customs union and two motions demanding a second referendum.
SNP Westminster leader Ian Blackford said his party would back the Boles motion, despite preferring the UK to remain in the EU.
"The key thing for us is that we stay in the European Union," Mr Blackford told the BBC. "We want to revoke Article 50. We want to put it back to the people in a people's vote.
"But we will be prepared to compromise on the basis of protecting jobs, staying in the single market and customs union, so we will vote for the Boles amendment."
Other Brexit options tabled as MPs took control of the Commons agenda for a second time include no deal, membership of Efta and the EEA without a customs union and a power for MPs to vote to block a no-deal Brexit.
Only those selected by Speaker John Bercow will be put to a vote.
No proposal won a majority in last week's votes, but if a consensus emerges on Monday MPs are expected to take control of the Commons timetable again on Wednesday in the hope of legislating for it.
Despite seeing her deal again defeated on Friday, the Prime Minister is determined to bring it back to the Commons for a fourth time in a final roll of the dice to get it over the line before the EU deadline of April 12.
Downing Street sources confirmed that the Government will whip against the business motion enabling Monday's indicative votes to go ahead.
Mrs May has previously warned that the unusual procedure risks creating a harmful constitutional precedent.
Some in Parliament believe that if MPs begin to gather around a "softer" Brexit, it may finally convince Brexiteer hold-outs that the PM's deal now represents the "hardest" break with the EU available.
But Mrs May's hopes of winning the support of her parliamentary allies in the Democratic Unionist Party were looking slim.
The Northern Irish party's Brexit spokesman Sammy Wilson said it will continue to vote against the PM's withdrawal deal, as it would "take us away from the country that we fought to stay part of".
And Tory rebel Richard Drax, who backed the deal last week, said he made "the wrong call" and would now oppose it, announcing his move in the Commons and demanding that the Prime Minister should resign if she is not prepared to take the UK out of the EU on April 12.
The pro-EU Justice Secretary David Gauke has warned Mrs May not to ignore the will of Parliament if it does swing behind a "softer" deal, such as Mr Clarke's customs union plan, which came closest to winning a majority last week.
And in an interview with the BBC, Tory chief whip Julian Smith said that the parliamentary arithmetic means a "softer type of Brexit" is inevitable.
But the Prime Minister has set her face firmly against a customs union, warning it runs counter to the Conservatives' election manifesto and would inhibit Britain's ability to strike trade deals around the world.
If she were to give way, she would provoke a furious reaction from Brexiteers, with International Development Secretary Penny Mordaunt and Transport Secretary Chris Grayling among the ministers reportedly ready to resign.
In an open letter to MPs, the chief executive of Siemens UK urged them to unite around a "customs union compromise" which he said was "essential" to frictionless trade and would save companies billions each year.
Juergen Maier warned that the UK used to be a "beacon for stability", but is now becoming a "laughing stock".
But Chief Secretary to the Treasury Liz Truss told BBC Radio 4's Today programme: "It's not clear to me that going softer is the way to command support."
She added: "If you look at the parliamentary arithmetic now, it's not clear that something like a customs union actually commands support."
Ms Truss said: "I think that we are well prepared for no deal. I don't have any fear of no deal."
And prominent Brexiteer Jacob Rees-Mogg told LBC he was worried that Mrs May was "more concerned to avoid a no-deal Brexit than anything else" and might "decide to go for a customs union tacked on to her deal".
Mr Rees-Mogg said Mrs May's Withdrawal Agreement might have won MPs' approval on Friday, when it was rejected by 58 votes, if the alternative had been a general election.
A letter to the Prime Minister signed by 170 Tory MPs demanded Britain leave by May 22 at the latest "without or without a deal".
In a sign of her waning authority, it was written by Brexit minister Chris Heaton-Harris, according to The Times.