Theresa May could be forced to accept a soft Brexit in order to end the deadlock if MPs unite around proposals for a customs union, a cabinet minister has warned.

David Gauke, the justice secretary, said the country had been plunged into “national crisis” and fired off a warning to Tory Brexiteers that the party may have to accept a “second or third choice” as it does not have the numbers to push its manifesto pledges through parliament.

As MPs prepared to seize control of the Commons agenda for a fresh round of indicative votes, Mr Gauke said the prime minister would have to “look very closely” if parliament voted in favour of a customs union.

The prime minister is facing pressure from all sides heading into another momentous week, with speculation mounting that she could be driven to call a snap general election if she fails in her fourth attempt to get her Brexit deal signed off by MPs.

Downing Street has made clear that she intends to bring it back to the Commons for a fourth vote – possibly on Tuesday or Wednesday.

Mr Gauke said her deal was still the best option but Ms May must consider a plan by veteran Tory Ken Clarke for a customs union – which came closest to securing a majority in the last round of votes – if MPs back it.

“I think she would need to look very closely at that,” he told the BBC’s Andrew Marr Show.

“If parliament is voting overwhelmingly against leaving the European Union without a deal but is voting in favour of a softer Brexit, then I don’t think it’s sustainable to ignore parliament’s position and therefore leave without a deal.”

His comments are likely to infuriate Brexiteers, who believe a customs union does not deliver on the promise of the referendum. Ms May has also repeatedly rejected the idea.

Mr Gauke said: “I think we also have to recognise that my party does not have the votes to get its manifesto position through the House of Commons at the moment.

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“I think the prime minister’s deal does reflect the manifesto position but we, as you say, have been defeated three times.

“So we do then need to come back and use our judgement as to what is the best way forward.

“Consider what the options are and I do think we’re in an environment where it’s not about just going for your first choice, sometimes you do have to accept your second or third choice in order to avoid an outcome that you consider to be even worse and MPs are entitled to use their judgement as to what the real choices are available for us as a country and come to what they consider to be the best conclusion.”

Mr Gauke restated his opposition to a no-deal exit, saying he could no longer remain part of a government that pursued such a strategy.

His comments set him at odds with many on the Brexiteer wing, with some 170 Tories, including several cabinet ministers writing to the prime minister to urge her to take the UK out of the EU as soon as possible.

Conservative Party deputy chairman James Cleverly insisted the party was not gearing up for a snap general election to resolve the Brexit deadlock but admitted it was doing some “sensible and pragmatic” planning.

“I don’t think an election would solve anything. Time is of the essence, we have got Brexit to deliver. We don’t want to add any more unnecessary delay,” he told Sky News’s Sophy Ridge On Sunday programme.

“We have got a minority government in a turbulent time,” he added.

He said it was an “inevitable possibility” that Ms May would lead the party into a snap election, despite a raft of blatant leadership bids from senior Tories such as Dominic Raab and Liz Truss.

Meanwhile, former Tory prime minister Sir John Major warned that if parliament was unable to agree a way forward there could come a point where there would have to be a government of national unity.