Theresa May could win parliament’s approval for her controversial Brexit deal in return for guaranteeing another referendum, under a new plan being drawn up by a cross-party group of MPs. The new vote would give the British people a simple choice: to confirm the decision or stay in the EU.

The initiative, aimed at breaking the political impasse, is being advanced by Labour MPs Peter Kyle and Phil Wilson and has won the support of prominent Remainers in the Tory party including Sarah Wollaston, Dominic Grieve and Anna Soubry.

Kyle says the idea, which is likely to be put forward as an amendment to the EU withdrawal bill, is also being taken seriously by “people at a high level in government” as a potential way to resolve the Brexit crisis.

The amendment would offer all MPs the chance to support, or abstain on, the withdrawal bill and would specify that, if passed, the decision would be implemented on the condition it was put to the public for approval in a second referendum.

If the amendment passed through parliament but the deal was rejected in the subsequent referendum, the UK would stay in the EU under current arrangements.

If, however, the British people confirmed the decision of MPs to leave the EU under the terms of May’s deal, Brexit on these terms would immediately come into effect without any need for it to return to parliament.

“The beauty of this plan is that it holds attractions for both Leavers and Remainers. For Leavers, if the deal is confirmed by the British people, it offers a definitive end to the withdrawal process with Brexit sealed once and for all. For Remainers, on the other hand, it offers the chance to make the case to stay in the EU to the public, based on facts not promises as before,” said Kyle.

“Remainers could vote for May’s deal, or abstain, even though they might not like it, in the knowledge that they could campaign against it later in the referendum.

“The key is that Labour and Tory MPs could choose their own way of allowing the bill to pass, yet both will still be able to campaign in the referendum for their real objectives and their principles will remain intact,” he added.

Further efforts could be made to wrest control of the Brexit process from the prime minister. Photograph: PA

Kyle said the plan should also appeal to Theresa May as the only way to get her deal approved both by the factionalised House of Commons and the British public. May’s deal was rejected by a margin of 230 votes, with 118 Tory MPs opposed.

Together with Commons clerks, the two Labour MPs have found a precedent for the plan in the way the Good Friday Agreement on Northern Ireland power-sharing passed through parliament in 1998.

Tory MP Sarah Wollaston, who has been considering when to table an amendment in favour of a second referendum, said the plan was a “very sensible way forward that I would be happy to support”. Soubry and Grieve welcomed the move, as they had always believed that any Brexit deal should be subject to a second referendum.

Further efforts could also be made this week by MPs to wrest control of the Brexit process from Theresa May, allowing the 29 March exit day to be delayed, after attempts to do so were rejected less than a fortnight ago. An amendment by Labour’s Yvette Cooper would suspend parliamentary rules which say that only the government can propose legislation to delay article 50, and put MPs in charge, giving them three days in control.

Labour leader Jeremy Corbyn, who has been criticised by some MPs in his party for appearing too ready to help the Tories deliver Brexit, said the option of Labour throwing its weight behind a second referendum would remain “on the table” if the prime minister refused to agree to Labour’s Brexit proposals. “We will keep all options on the table – as agreed in our conference motion – including the option of a public vote,” Corbyn said.

Kyle said that his plan had three key advantages. First, it would be a way of reconciling the country to whatever the eventual Brexit outcome turns out to be because it would deliver the “double lock” consent of both MPs and the public. Second, it would offer a definitive end to the Brexit process. And third, it would break the parliamentary paralysis that has long dogged Brexit.