Soft Brexit cabinet ministers are quietly backing a parliamentary bid to make it legally impossible for the UK to leave the EU without a deal on 29 March, the Conservative MP behind the bid has claimed.

The bill tabled by the former minister Nick Boles would force the government to delay the UK departure from the EU unless a consensus could be found by early March. That would in effect make it impossible for the government to legally leave the EU without a deal on 29 March.

“This bill is about creating the space for a compromise by ruling out a no-deal Brexit,” Boles told the BBC. “We have had indications that many ministers, including cabinet ministers, are very, very keen to see it pass and are telling the prime minister that they will not vote against it.”

However, to force the government to grant time for the bill, Boles is set to table an amendment to the government’s Brexit motion that MPs will vote on on 29 January. The amendment would give parliamentary time for Boles’ bill, taking precedence over government business.

If the amendment does not pass, the government would otherwise be unlikely to allow time for Boles’ bill. Using that new power, Boles and his allies hope to then give MPs a vote on his European Union withdrawal number 2 bill.

The plan has attracted the backing of a cross-party coalition of MPs, including Labour’s Yvette Cooper, Liz Kendall and Hilary Benn, the Conservative Sir Oliver Letwin and the Lib Dem Norman Lamb.

Lamb said the MPs had been forced to consider radical action. “We’re in unprecedented circumstances now, highly dangerous times for the country,” he said.

“It’s critically important that parliament is able to decide if we drift into a no-deal situation. There’s no indication from the prime minister she recognises that immense risk. But there are many in the cabinet who do understand that massive risk.”

MPs believe the plan would carry far more weight that amendments to a government motion, which would be politically but not legally binding. In contrast, Boles’ amendment would make it legally binding for the government to avoid a cliff-edge Brexit by mandating an extension of article 50 if parliament cannot agree a way forward.

Its supporters hope it will attract support from a wide range of factions, including Tories who would like to see a looser free trade deal as well as MPs who back a second referendum because it would give adequate time for a new direction to be forged.

It remains uncertain, however, if the EU27 would permit an extension unless there were a concrete reason such as formal ratification of the deal, a second referendum or an election, rather than just extra negotiating time.

Boles said there was now “a bandwagon rolling” behind the plan. “I very much hope that any MP who shares my view that a no-deal Brexit would be a disaster will jump on board,” he said. “I have been told directly by ministers not in the cabinet that they have said that they would resign if they are whipped to vote against it.”

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Cabinet ministers have remained publicly mute on the subject. However, speaking on a conference call with business leaders alongside the business secretary, Greg Clark, and the Brexit secretary, Stephen Barclay, the chancellor, Philip Hammond, admitted that the government was “not in control” of the process and suggested Boles’ amendment could lead to article 50 being “rescinded”, according to a transcript obtained by the Daily Telegraph.

Boles said the transcript of the call “made quite plain that [Hammond] thought this was fantastic”.

Boles had originally intended for parliament’s liaison committee of select committee chairs, headed by the Tory MP Sarah Wollaston, to take submissions from MPs on the future of the Brexit process, but that aspect has now been shelved because of objections from the committee.

The MP tweeted on Thursday that the liaison committee was “not keen to take the role that is proposed for it”. Boles said that part had been an “optional extra” and the focus was taking no deal off the table.