The UK may need a cross-party commission like the one set up to decide on new powers for Scotland after the 2014 independence referendum, one of the MPs behind a last-ditch effort to stop a no-deal Brexit has said.
Yvette Cooper said that if MPs cannot reach a consensus on the terms the UK should leave the EU under in the next two days, the government should seek a long extension to allow a process that could break the deadlock.
Ms Cooper has joined with Tory MP Sir Oliver Letwin - who led the process of indicatives votes that failed to produce a clear Brexit alternative - to propose emergency legislation to stop a no-deal scenario.
The MPs have tabled a bill requiring the Prime Minister to extend the negotiation process beyond April 12.
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Following the failure of MPs to unite behind an alternative to Mrs May's plan on Monday, the group aims to pass the bill through the Commons on Thursday.
Number 10 has made clear on Tuesday that exit without agreement in 10 days time remains the legal default unless MPs approve a deal.
And EU chief negotiator Michel Barnier said a no-deal departure was becoming "day after day more likely".
A leaked letter from cabinet secretary Sir Mark Sedwill, obtained by the Daily Mail, showed that the UK's top civil servant has warned of 10% food price hikes, economic recession and disruption to security if Britain crashes out without a deal.
Instead of initiating a third round of indicative votes on Wednesday, when Parliament once more has control over the Commons timetable, Sir Oliver will table a paving motion for approval by MPs which would allow debate and votes on Ms Cooper's bill on Thursday. Indicative votes could then take place.
The single-clause bill requires the Prime Minister to table her own motion seeking MPs' approval for an extension to the Article 50 process of Brexit talks to a date of her choosing.
The group behind the bill, which also includes former Tory chair Dame Caroline Spelman, Commons Brexit Committee chair Hilary Benn, former attorney general Dominic Grieve and Liberal Democrat Norman Lamb, aims to push it through all of its Commons stages on Thursday, and hopes the House of Lords would then grant its approval ahead of an emergency EU summit on April 10.
Ms Cooper said: "We are now in a really dangerous situation with a serious and growing risk of no-deal in 10 days' time. The Prime Minister has a responsibility to prevent that happening.
"She needs to put forward a proposal, including saying how long an extension she thinks we need to sort things out.
"If the Government won't act urgently, then Parliament has a responsibility to try to ensure that happens even though we are right up against the deadline."
Sir Oliver added: "This is a last-ditch attempt to prevent our country being exposed to the risks inherent in a no-deal exit.
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"We realise this is difficult. But it is definitely worth trying."
MPs rejected a call for a customs union with the EU by just three votes on Monday, while a demand for a second referendum was defeated by 12 and a "Common Market 2.0" deal by 21.
Number 10 officials are understood to have indicated that if the Prime Minister could not get her Withdrawal Agreement through on a fourth vote then she would seek a longer extension to the Brexit process "possibly until the end of the year".
But she is believed to favour a no-deal Brexit rather than the possibility of revoking Article 50 if the choice came to it.