Theresa May’s hopes of winning parliamentary approval for her Brexit deal grew last night as the leader of Conservative backbenchers softened his opposition – and suggested he will recommend that MPs back it – if the prime minister secures new assurances on the Irish backstop.

The move by Sir Graham Brady, who chairs the 1922 committee, is a significant shift and came as senior government sources predicted that May could hold the crucial meaningful vote as early as this week. Only weeks ago Brady successfully championed a Commons amendment that instructed May to return to Brussels to negotiate the complete removal of the backstop from the withdrawal agreement and find another way to solve the Irish border issue.

The Brady amendment, which passed through the Commons by 318 votes to 310 on 29 January, called for “the Northern Ireland backstop to be replaced with alternative arrangements to avoid a hard border” and said a deal on leaving the EU would be dependent on its being removed.

But Brady and other leading Tory backbenchers – who fear the alternative to May’s deal would be a no-deal Brexit, a lengthy delay or a second referendum – now seem prepared to drop their demand on condition that they win reassurances that the backstop will be temporary and does not risk locking the UK into a permanent EU customs union.

Brady told the Observer last night that his intention had always been to “ensure that the backstop could not assume a permanent status, trapping the UK in the EU customs union”. He added: “As long as the attorney general [Sir Geoffrey Cox] is able to assure the House that he has a legally binding guarantee that the backstop can only be temporary, I would accept that and would urge others to accept it.”

May’s decision last week to hold a vote on delaying Brexit if her deal is rejected again, coupled with Labour’s coming out more clearly in support of a second referendum, have been key factors in persuading many Conservatives to consider switching their vote to support the prime minister. Other senior Tories say that while May still faces an uphill struggle to win over many of the most hardline Tory opponents of her deal – and the 10 MPs from the Democratic Unionist Party – many others who helped inflict the 230-vote defeat in January are now looking for ways to back down and support it.

A decisive test for the prime minister, according to senior Tories, will be whether the DUP accept any changes as sufficiently binding. “If the DUP can take comfort, most of our MPs will be happy and take their lead from them. If that happens I really think she has a good chance,” said one former Tory minister.

Sir Graham Brady has said he will support Theresa May’s deal it the Irish border backstop is time limited. Photograph: Leon Neal/Getty Images

The Tory whips are also desperate to win support from Labour MPs in leave-supporting areas. Details of a package of measures on boosting workers’ rights and a fund for deprived areas is expected to be announced by ministers this week

Cox held talks in Brussels last week with senior EU officials on whether an additional appendix or codicil could be drawn up stating that no one intends the backstop to be permanent. He is expected to return for more talks this week.

In an interview with the German newspaper Die Welt, EU chief negotiator Michel Barnier said he was looking at drafting a statement as an adjunct to the withdrawal agreement – but insisted that Brussels would not put a time limit on the backstop – as demanded by many Tory MPs.

“We will not allow a time limit or a one-sided exit right,” Barnier said. “What can exist is the commitment to limit the backstop through an agreement on the future relationship … in the form of an interpretive document, like the joint letter from Donald Tusk and Jean-Claude Juncker. If this document were combined with a written commitment from the British, it would have a much greater power.”

In an interview with European newspapers yesterday, Barnier suggested that even if May were to get her deal through in mid-March, she would still need to delay Brexit beyond 29 March in order to pass the necessary legislation.

Another senior Tory, Robert Halfon, chair of the education select committee, said he would support May’s deal having voted against in January. “We face economic calamity if there is no deal or political calamity if there is a second referendum, which would destroy trust in our democracy.”

Friends of former foreign secretary Boris Johnson said he would need to see details of an “escape route or freedom clause” from the backstop before he could consider “holding his nose” and backing the deal.

Jacob Rees-Mogg, who as chair of the European Research Group represents the 50 or so Tories most in favour of a hard Brexit, conceded last week that he, too, was shifting. Only last month, during a speech in Westminster, he said: “As long as the backstop is there, I will not vote for the deal.” However in a BBC Radio 4 interview last week, he said he would be prepared to accept changes which guarantee that the backstop would be time-limited, rather than insisting on its removal altogether.

“A changed deal is a changed deal. Of course, I would be open to considering that,” he said.

* This article was amended on 4 March 2019 to correct references to the Ulster Unionists where the Democratic Unionist Party was meant.