A cross-party group of MPs has put forward a bill to prevent a no-deal Brexit in 10 days' time.
If passed into law, the bill would require the PM to ask for an extension of Article 50 - which mandates the UK's exit from the EU - beyond the current 12 April deadline.
Labour MP Yvette Cooper presented the bill - which supporters hope they can pass through the Commons in one day.
The prime minister is expected to make a statement shortly.
It comes after the cabinet, which remains split over Brexit, met for eight hours in No 10.
The BBC's John Pienaar said Theresa May's ministers considered plans to "ramp up" no-deal Brexit preparations and a snap general election was also discussed.
Ms Cooper's bill would make it UK law for the PM to ask for an extension to prevent a no-deal, but it would be up to the EU to grant it - or not.
In March, MPs voted against leaving the EU without a deal, but it was not legally binding.
Meanwhile, the EU's chief negotiator has said a no-deal Brexit is now more likely but can still be avoided.
Michel Barnier said a long extension to the UK's 12 April exit date had "significant risks for the EU" and a "strong justification would be needed".
France's President Emmanuel Macron and Irish Taoiseach Leo Varadkar are meeting in Paris to discuss the impact of Brexit.
President Macron told reporters that the EU "cannot be hostage to the political crisis in the UK", and the government must come forward with "credible" reasons for an extension.
He said these could include an election, second referendum, or alternative proposals for the future relationship, such as a customs union.
Mr Varadkar said the UK was "consumed by Brexit", but the EU should not be.
He said the EU "needs to be open" about any proposals the UK brings, including a longer extension, and they will do what they can to "assist".
But he added: "We gave the UK some time, some space and some opportunity to come up with a way forward... [but] as things stand, they will leave on 12 April without a deal."
Tory MP Sir Oliver Letwin, who supports Ms Cooper's bill, said: "This is a last-ditch attempt to prevent our country being exposed to the risks inherent in a no-deal exit.
"We realise this is difficult. But it is definitely worth trying."
Ms Cooper said the UK was "in a very dangerous situation" and MPs "have a responsibility to make sure we don't end up with a catastrophic no deal".
Speaking to BBC Radio 4's World At One, she added: "We have been attempting to squeeze into just a couple of days a process that really should have been happening for the last two years - a process of trying to build a consensus around the best way forward.
"It is what the prime minister should be doing. It is the prime minister's responsibility to ensure we don't leave the country less safe."
Why is this bill unusual?
Normally the government chooses which bills to present to Parliament in order for them to become law.
But - much to the government's disapproval - MPs voted to allow backbenchers to take charge of business in the Commons on Wednesday.
This gives backbenchers the opportunity to table their own bills, such as this one from Yvette Cooper.
A copy of the bill shows that they want to push it through the commons in one day.
As the backbenchers will be in charge, they will also be able to vote to set aside more time on another day, if they need to complete the process or hold further indicative votes.
However, the bill would also have to be agreed by the House of Lords and receive Royal Assent before it became law - which if the Commons agrees it on Wednesday, could happen as soon as Thursday.
Brexiteer Tory Sir Bill Cash said trying to go through these stages in one day made it a "reprehensible procedure".
But Speaker John Bercow said that, while it was "an unusual state of affairs", it was "not as unprecedented as he supposes" - citing recent bills on Northern Ireland that have been passed at the same speed.
In the latest round of indicative votes on Monday, MPs voted on four alternatives to the PM's withdrawal deal, but none gained a majority.
MPs rejected a customs union with the EU by three votes. A motion for another referendum got the most votes in favour, but still lost.
The votes were not legally binding, but they had been billed as the moment when Parliament might finally compromise.
The Independent MP Chris Leslie tweeted that MPs would be seeking more time for indicative votes to take place on Monday.
With cross-party group,@TheIndGroup MPs are supporting amendments to both legislate for Article 50 extension (Commons time on Weds for this) & to provide for next round of indicative votes (on Monday 8th April) where obvious compromise will be #PeoplesVote & Customs Union combo✅ pic.twitter.com/bcl380Cspo— Chris Leslie (@ChrisLeslieMP) April 2, 2019
Liberal Democrat MP Norman Lamb said he is considering resigning the whip after his party refused to back proposals for a customs union and Common Market 2.0 on Monday.
He told BBC News: "If you are seen to be unreasonable, not engaging to find solutions, I don't think it is very attractive to the people."
Earlier, Mr Barnier said: "No deal was never our desire or intended scenario but the EU 27 is now prepared. It becomes day after day more likely."
Mrs May's plan for the UK's departure has been rejected by MPs three times.
Last week, Parliament took control of the process away from the government in order to hold a series of votes designed to find an alternative way forward.
Eight options were put to MPs, but none was able to command a majority, and on Monday night, a whittled-down four were rejected too.
- Tuesday 2 April: A five-hour cabinet meeting
- Wednesday 3 April: Potentially another round of indicative votes, and Yvette Cooper's bill to be debated
- Thursday 4 April: Theresa May could bring her withdrawal deal back to Parliament for a fourth vote, while MPs could also vote on Ms Cooper's bill
- Wednesday 10 April: Emergency summit of EU leaders to consider any UK request for further extension
- Friday 12 April: Brexit day, if UK does not seek / EU does not grant further delay
- 23-26 May: European Parliamentary elections