Labour has tabled a cross-party motion to try to stop a future prime minister pushing through a no-deal Brexit against the wishes of MPs.
The party plans to force a vote on Wednesday which would give MPs control of the timetable on 25 June.
Labour says if the motion passes, MPs will be able to introduce legislation on that day to avoid a no-deal scenario at the end of October.
Some Tory leadership hopefuls have said they would leave the EU without a deal.
For others, the prospect is unacceptable.
Leaving on a no-deal basis – without any agreement on the shape of the future relationship between the UK and EU – could lead to significant disruption.
The EU has previously said border checks would have to be brought in, affecting things like exports and travel and creating uncertainty around the rights of UK citizens living in the EU and vice-versa.
The government normally controls business in the Commons – but MPs have previously seized control to legislate in favour of extending the Brexit process.
Labour’s Brexit spokesman Sir Keir Starmer said MPs “cannot be bystanders” while the next prime minister “tries to crash the UK out of the European Union without a deal and without the consent of the British people”.
“That’s why we are taking this latest measure to end the uncertainty and protect communities across the country,” he said.
“My challenge to MPs who disagree either with a no deal Brexit or proroguing Parliament is to back this motion and act in the national interest.”
Leadership candidates Dominic Raab and Esther McVey have both said they would consider shutting down Parliament early – proroguing – in order to drive through no deal.
The motion has cross-party backing, including from one Tory MP – Oliver Letwin – who is supporting Michael Gove in the leadership contest.
It has been signed by Jeremy Corbyn, SNP Westminster leader Ian Blackford, Lib Dem leader Vince Cable, Plaid Cymru Westminster leader Liz Saville-Roberts and former Green Party leader Caroline Lucas.
Due to the confidence and supply agreement with the Democratic Unionist Party, the Tories have a majority in Parliament of five.
That means it would take only three Conservatives to vote with the Labour motion for it to pass – if all opposition party MPs back it.
International Trade Secretary Liam Fox told the BBC the move was “typical of the current Labour Party”, saying: “Jeremy [Corbyn] wants to be a Remainer in the south and a Leaver in the north.”
“Is it (the Labour Party) actually trying to block Brexit now?” he asked.
“If Labour tomorrow wants to make no deal impossible, they are making revocation [of Article 50] and staying in the EU a possibility.”
Justice Minister Robert Buckland called it “parliamentary game-playing”, asking: “What is the point of all this sound and fury?”
Analysis: By Jonathan Blake, BBC political correspondent
Another attempt to re-write the rules, another heave in the procedural tug of war, another day of drama in Parliament. But will it work?
It’s not a straight vote for or against a no-deal Brexit – that would not change the fact that it is written in law and agreed with the EU that Brexit will happen on 31 October.
Think of this plan not as a knockout blow in a boxing match, but the first of a complicated sequence of moves in a chess game.
Labour want to pull off something similar to what happened in March, when MPs took control of parliamentary time to force the government to request an extension to the Brexit process from the EU.
Step one is seizing control of business in the House of Commons, and that’s clearly the plan this time around.
Beyond that, the details aren’t clear.
Compelling the new prime minister to ask the EU to delay Brexit further is the most likely option. But the answer of course, might be “no”.
Ex-director of legislative affairs at No 10, Nikki da Costa called Labour’s move “extraordinary”.
Change UK MP Chris Leslie said Tory leadership candidates “who think they can just shut down the Commons” to deliver a no-deal Brexit should “wake up and realise there is a cross-party majority of MPs who just will not let this happen”.
And Labour MP Ben Bradshaw said all MPs who “profess to oppose prorogation” for no deal should back the motion.
“It could be our only insurance policy,” he added.
Can Parliament stop a no-deal Brexit?
The default position in law is that the UK will leave the EU on 31 October – and if nothing changes, Brexit will happen regardless of whether there is a deal or not.
MPs wanting to stop a new PM leaving without a deal do however have a number of options at their disposal.
One would be to pass legislation requiring the government to seek an extension to the UK’s membership. The EU would have to agree to an extension for it to be granted.
However this would first require MPs to seize control of the parliamentary agenda, as Labour is attempting.
Another would be to use a vote of no confidence to bring down a government committed to pursuing a no-deal exit.
MPs could also use motions or political pressure to try and force the government into changing course.
Labour’s move came after shadow chancellor John McDonnell insisted the party was not in “turmoil” over its Brexit stance.
The shadow chancellor was responding to questions about a heated Parliamentary Labour Party (PLP) meeting on Monday in which Mr Corbyn faced criticism from his MPs.
A number of MPs expressed concerns that it had now become “normalised” for Labour voters to back other parties over Brexit, while others strongly criticised Mr Corbyn’s handling of anti-Semitism.
Speaking at the Times CEO summit, Mr McDonnell said he welcomed lively debate.
“I would rather have people get up and say, ‘This is what I feel,’ passionately, rather than sneak away to the corners,” he said.
He said the party had promised to respect the referendum result in its manifesto and now, after failing to secure a deal with the Conservatives in cross-party talks, the situation meant “most probably going back to a public vote” – his preference being a general election.
But Labour MP Anna Turley said colleagues were “shell shocked” afterwards, such had been the level of anger expressed.
“I think the cleaners are probably still mopping up the blood,” she added.
Ms Turley, who did not attend Monday’s meeting herself, is calling for a “firmer position on Brexit” from the party leader – with her preference being for a confirmatory vote on any deal agreed between the UK and the EU.
She told the BBC’s Politics Live programme colleagues who had been present made clear it had been a “messy and difficult” meeting.
After the PLP meeting, a spokesman for Mr Corbyn said: “The PLP is very passionate about lots of issues not just about Brexit. That’s what we would expect.”