Theresa May has announced to the Commons that she plans to return to Brussels and seek to reopen the painstakingly negotiated withdrawal agreement so as to seek legally-binding changes to the Irish backstop.

The prime minister confirmed that the government would back an amendment tabled by senior Tory backbencher Sir Graham Brady, which seeks to replace the Irish backstop with “alternative arrangements”.

“This amendment will give the mandate I need to negotiate with Brussels an arrangement that commands a majority in this house,” she said, denying charges from some MPs that seeking to reopen a deal the EU has repeatedly insisted is final was to pursue “fantasies”.

“What I’m talking about is not a further exchange of letters, but a significant and legally-binding change to the withdrawal agreement,” May said. “Negotiating such a change will not be easy. It will involve reopening the withdrawal agreement, a move for which I know there is limited appetite among our European partners.”

May also confirmed she would most likely back the so-called “Malthouse compromise” – a proposal to secure backstop-free departure thrashed out by the housing minister Kit Malthouse together with both Brexit and remain-backing MPs.

Answering an intervention from the Conservative MP Nicky Morgan, one of those involved in the plan, May called it “a serious proposal that we’re engaging with sincerely and positively”. The plan was finalised too late to be considered as a formal amendment.

Noting that MPs had voted down her initial deal and made it plain they did not want a no-deal departure or an election, May said: “So the world knows what this house does not want. Today we need to send an emphatic message about what we do want.”

May said that if a deal were not made by 13 February, the government would table another statement setting out what the government planned to do next, which MPs would be able to amend, giving them another opportunity to reject a no-deal Brexit.

Before May spoke the Speaker, John Bercow, said he had selected Brady’s amendment among seven to be voted on at 7pm, as well as one from the Tory MP Dominic Grieve seeking to push the government to hold indicative votes on the way forward, and the Labour MP Yvette Cooper’s proposal to stop no deal.

Full list: amendments to May’s statement on defeat of her Brexit bill Read more

The prime minister said these two proposals were “deeply misguided”, and had significant consequences.

“Both amendments seek to create and exploit mechanisms that allow parliament to usurp the proper role of the executive,” she said.

The other amendments accepted were an official Labour amendment, which calls for a customs union; an SNP plan seeking an extension to article 50; a cross-party plan to extend article 50 and another cross-party amendment advising against no deal.

Jeremy Corbyn confirmed that as well as the frontbench amendment, the party would also formally support Cooper’s plan – but would seek to have the potential one-year extension of article 50 reduced to three months.

This would be “the opportunity to take ‘no deal’ off the table”, the Labour leader said in a speech during which he was repeatedly interrupted by shouts and points of order by Conservative MPs who felt he was ignoring interventions.

The real “obstacle to a solution” was May and her government, Corbyn said. “They spent most of the last two years arguing amongst themselves rather than negotiating with the EU, and still they’re arguing amongst themselves and failing to come up with a workable solution,” the Labour leader said. “Tonight I hope this house does its job, and leads where this government has failed.”

Downing Street had previously insisted that reopening the 585-page withdrawal agreement was unacceptable to the EU27, and risked other governments trying to push for fresh concessions on other issues, such as fisheries or the status of Gibraltar.

Earlier, Jacob Rees-Mogg, the chair of the backbench European Research Group, and the former foreign secretary Boris Johnson, had both demanded a clear statement from May that she was willing to reopen the fraught issue of the backstop.

“We need to go back into the text of the treaty and solve the problem. That is the way to unite remainers and leavers in the Conservative party and across the country,” Johnson said.

Boris Johnson (@BorisJohnson) We can’t have some codicil or letter or joint declaration. We need to go back into the text of the treaty and solve the problem. That is the way to unite remainers and leavers in the Conservative party and across the country 2/2

The prime minister is expected to hold calls with EU leaders later on Tuesday, to clear the way for another round of negotiations.

Senior Conservatives anxious that the government is drifting towards a no-deal Brexit, including the business minister Richard Harrington, had demanded that May set a two-week deadline for securing a settlement.