An amendment proposing a second referendum from two backbench Labour MPs will not now be put to a vote when Theresa May brings her Brexit deal back to parliament next week.

Labour’s leadership wants attention on Tuesday to focus on May’s deal – but the party has not ruled out a second referendum motion later in the week if the prime minister fails to win MPs’ backing.

Campaigners for a second referendum believe they can only a win a majority in the Commons if it is seen as the sole option to break the deadlock preventing any Brexit deal passing through parliament.

Writing in the Guardian, the former Labour communications chief Alastair Campbell, who is involved with the People’s Vote campaign, said that Tuesday “must belong to Mrs May being made to see her deal will not – and cannot – fly” and called on the backbenchers to hold fire.

Labour MPs Peter Kyle and Phil Wilson had been pushing an amendment that would allow May’s Brexit deal to pass through the Commons on condition that its approval was subject to a second referendum.

They believed their offer could be attractive to some Conservatives because it would at least allow the Brexit deal to pass through parliament, thus ending what could otherwise be months and months of political wrangling.

“I hope they will not push their amendment to a vote on Tuesday,” Campbell wrote. “I think the public needs to see very clearly that even though the ticking clock may have reduced the majority against Mrs May’s deal from the stratospheric level 230 of January, it continues to lack the parliamentary support needed to go through.”

Kyle indicated he was minded to follow Campbell’s advice without being too precise about the timetable he would follow. “Our compromise deal will be pushed when we judge other options have been exhausted and MPs are ready for compromise and a creative way out of this mess,” he said.

Labour sources said the party wanted to have a “clean vote” on May’s deal on Tuesday, and that any amendment the party puts down in the name of Jeremy Corbyn would be designed “not to get in the way”.

With little sign of a breakthrough, it is not clear how May will get her deal approved on Tuesday. However, the opposition wants to keep its options open for later in the week if May fails. The prime minister has promised a vote on whether the UK should leave on 29 March with no deal, and if that is defeated, as is expected, a second vote would follow on whether the UK should delay its exit from the EU for further negotiations.

Labour is considering whether to put its own second referendum amendment to the Brexit delay vote.

Significantly, some backbenchers who had been sceptical about the idea of a referendum believe that resistance to the idea is fading. They say Labour MPs from pro-leave seats are coming under increasing pressure to back what is now party policy.

Sir Keir Starmer will address a meeting of MPs from the Labour Leave Seats group on Monday, accompanied by the pollster Peter Kellner and Kyle and Wilson. Jenny Chapman, from Starmer’s team, who had previously been regarded by colleagues as wary about a referendum, is also expected to make the case for a fresh poll.

One backbencher from the group said: “If you’d asked me last week, was there a majority for a second referendum, I would have said no way. Next week, if it does fail, it will be by a handful of votes.”

They added that the number of MPs willing to defy the whip and vote against a referendum might be just 15 or so Labour MPs – much lower than previously thought. Others had put the estimate at 25 rebels or more.