The cabinet minister in charge of Brexit has held detailed talks with Labour MPs who are championing plans for a second referendum – amid signs of mounting desperation inside Theresa May’s government about what to do if the prime minister’s deal suffers another crushing defeat on Tuesday.

Stephen Barclay, the Brexit secretary, called the meeting with Labour’s Peter Kyle and Phil Wilson in Downing Street last Thursday as negotiations with Brussels to resolve the deadlock over the Northern Ireland backstop floundered and ministers privately began to concede that May’s plan could be doomed.

Kyle told the Observer on Saturday that Barclay had “remained loyal to government policy”, which is to oppose any second referendum. But the MP for Hove said Barclay talked to him and Wilson for at least 45 minutes and was “fully engaged”.

Under the Kyle-Wilson plan, which could receive Labour’s official backing after Tuesday’s vote but before Brexit day on 29 March, an amendment would be put down in parliament allowing MPs to approve May’s deal, but on the condition that it is then put to the country in a second public vote. The other option on the ballot paper would be to remain in the EU.

Kyle said: “Clearly the Brexit secretary is fully engaged in the battle to deliver support for the vote on Tuesday. But that didn’t stop him from engaging fully with the possibilities our compromise offers. We didn’t enter the meeting with expectations of converting him to our cause but we hope that government now understands what our plan offers, should a new direction be demanded by parliament in the coming days.”

A senior Downing Street source said May remained “100% opposed to a second referendum, with bells on” and insisted that she and all cabinet ministers were still determined to persuade enough Tory MPs and the 10 DUP members to rally behind her deal this week.

But – with little sign of progress in talks with Brussels – there is a growing recognition among ministers that all other options must now be considered to avoid a disastrous no-deal outcome. Backers of the Kyle-Wilson plan, including senior members of the shadow cabinet, accept that there is currently no parliamentary majority for a second referendum.

But they believe this could change if May’s deal goes down and no alternative to it can win sufficient parliamentary backing. In that event, they say that even the prime minister could see the merits of fighting for her deal out in the country.

A Downing Street source said on Saturday that the RAF plane that ferries the prime minister and senior ministers to EU capitals would be on standby this weekend in case further last-minute meetings with Europeanleaders in Brussels or elsewhere were called. Negotiations were continuing “at all levels” before Tuesday’s vote. “These are tough talks with the [European] commission which we are expecting to go right down to the wire,” said the source. “The PM, ministers and her negotiating team are intensely focused this weekend on making progress so that ultimately we can, in the country’s best interests, leave the EU with a deal.”

If May loses on Tuesday, she has said she will call a vote on Wednesday on whether parliament should rule out a no-deal Brexit and then a further one, probably on Thursday, on delaying Brexit. No decision has yet been made on whether Tory MPs will be granted free votes on the two amendments, and Downing Street has refused to say how May will herself vote on them.

It is understood that if she loses, the prime minister has not ruled out trying to bring back her deal to parliament a third time, nearer to Brexit day, when she would tell MPs that the only alternative to backing her would be a lengthy delay that could mean Brexit not happening at all.

Labour will delay putting down an amendment in favour of a second referendum until nearer Brexit day. Campaigners for a second public vote now believe that their best chance of success will be late in the process when there is no other viable option. This could even mean tabling the plans after 29 March, assuming Brexit has been delayed.

But the new Independent Group of MPs will table an amendment for a second referendum to take place after a series of indicative votes on alternatives to May’s deal had been held.

At least 40 Labour MPs – and all but about nine Tories – are said to be opposed to a second referendum, although party sources believe this would change if Jeremy Corbyn came out strongly in favour. Writing on theguardian.com, the Labour MP Jon Cruddas and the director of Hope not Hate, Nick Lowles, argue that another referendum will only be seen as “legitimate” if the option of no-deal is also put on the ballot paper.

A poll by BMG for the anti-Brexit youth campaigns Our Future Our Choice and For Our Future’s Sake (details on theguardian.com) shows that some 87% of young people who were ineligible to vote in the 2016 referendum, but would now be old enough to do so, and would definitely take part, would back Remain. Some two million young people who could not vote last time due to their age would now be able to do so.