The Speaker, John Bercow, has allowed MPs to debate an amendment that could force Theresa May to present a new Brexit plan within three days if her current proposal is voted down, a decision likely to enrage the government.

Bercow has selected an amendment led by the Conservative MP Dominic Grieve to the business motion that sets up the latest five-day debate on the prime minister’s plan, which will culminate in a crunch vote next Tuesday.

The news comes as Labour edged closer to promising a near-immediate vote of no-confidence in the government if May loses the vote.

The amendment says that following defeat of the government’s plan, which is widely anticipated, “a minister of the crown shall table within three sitting days a motion … considering the process of exiting the European Union under article 50”.

Grieve said he understood his amendment had been accepted. “I believe it has,” he said, adding that it was “eminently sensible”. He said: “This solidifies and emphasises the key role of parliament.”

Other MPs who have signed the amendment include the former Tory cabinet minister Sir Oliver Letwin and ex-Tory ministers Jo Johnson, Guto Bebb and Sam Gyimah. It has also been backed by Labour MPs including Stephen Doughty and Chris Leslie.

The government believed such a business motion should be amendable only by a minister, and was expected to fiercely resist Bercow’s decision to accept the Grieve amendment.

Adding to the government’s likely concerns, a previous amendment by Grieve, voted through by the Commons before Christmas, means any statement that follows a defeat is in itself amendable – allowing MPs to put forward their own alternatives for the future of the Brexit process.

So if May came back with an alternative plan, that could be amended by MPs, giving parliament far more scope in setting the direction.

Sarah Wollaston, the Conservative chair of the health select committee, who also signed the amendment, said the aim was to prevent the government “running down the clock” towards no deal. Previously, the Commons had mandated the government to make a statement within 21 days.

“If and when the PM’s plan is voted down on Tuesday, MPs can’t be made to wait potentially until 12 February for the next vote. The situation is too urgent now,” Leslie said.

Also on Wednesday, two Labour frontbenchers indicated their party could soon push for a formal no-confidence motion in May’s government, which could trigger a general election if it was won.

Discussing the likely defeat of the Brexit plan next week, Barry Gardiner, the shadow international trade secretary, told BBC Radio 4’s Today programme: “Obviously, the next thing to do immediately after that is for there to be a vote of confidence in the government.”

Speaking later on Sky News, Andrew Gwynne, the shadow communities secretary, confirmed the plan. Asked if this meant a confidence vote would come “next week, basically” if May lost the vote. Gwynne replied: “Absolutely.”

It comes a day after the government suffered a significant defeat on the finance bill, in which a powerful cross-party group of MPs led by Labour’s Yvette Cooper passed an amendment limiting the government’s tax administration powers in the event of the UK leaving the EU with no deal, unless this was explicitly endorsed by MPs.

Those behind the amendment’s success conceded it might have little material effect on no-deal preparations. Instead, its purpose had been to galvanise MPs across the House of Commons and proves there was a parliamentary majority to oppose no deal.

The rebels said they could seek to amend any and every piece of legislation the government brings to parliament between now and March.

The prime minister is still seeking a form of legally binding reassurance from the European Union about the temporary nature of the backstop in the withdrawal agreement, the main bone of contention that has led Tory MPs to reject her deal.

David Lidington, May’s de facto deputy, said on Wednesday MPs should not expect Brussels to unpick the withdrawal agreement. “I don’t think that the British public are served by fantasies about magical alternative deals that are somehow going to sort of spring out of a cupboard in Brussels,” he told BBC Radio 4’s Today programme.