The vote on the withdrawal agreement bill will be make or break for Theresa May’s future as prime minister, Downing Street has indicated, as a member of her cabinet said defeat could also kill off the deal entirely.

No 10 said the key piece of Brexit legislation would be voted on in the week beginning 3 June, and talks with Labour would continue in the meantime.

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May is set for a tense meeting on Thursday morning with the 1922 Committee executive, the body that represents backbench Tory MPs, who have demanded she set out a timetable for her departure regardless of whether the bill is passed.

Sources on the executive said setting the date for the bill would not satisfy the demand from the 1922 Committee, whose more bellicose members have suggested radically rewriting party rules to force the prime minister from office.

“It might be her aim [to satisfy demands for a departure date] but it is not going to work. That is not the way we see it,” one source on the committee said. “The same demand still stands tomorrow, very much so. I don’t think she will set it out, but we are expecting her to do so. If the bill is defeated, that really is the end of the road though, even if can-kicking is a professional sport nowadays.”

Changing the rules was rejected by the majority of the committee but only on the condition that the prime minister makes the date of her exit more explicit. May will face a no-confidence vote from party officials and members on 15 June at an extraordinary general meeting – though it is non-binding.

May’s spokesman declined to confirm that the prime minister would see it as the last act of her premiership if she were to lose the vote at the bill’s second reading, but said the PM understood its significance.

“Clearly the significance of this piece of legislation can’t and I suspect won’t be underestimated,” the spokesman said. “The absolute determination and focus of the government will be to get this thing through and passed.”

Stephen Barclay, the Brexit secretary, speaking to the Lords EU select committee, suggested the vote would be the final throw of the dice for the deal negotiated with the EU’s Michel Barnier.

“I think if the House of Commons does not approve the [bill] then the Barnier deal is dead in that form and I think the house will have to then address a much more fundamental question between whether it will pursue … a no-deal option or whether it will revoke,” he said.

May faces a punishing week when the Commons returns from recess, including the vote on the withdrawal bill, a three-day visit from Donald Trump, D-day commemorations with world leaders in Portsmouth and France, as well as the byelection in Peterborough, a tightly contested seat being fought for by the Tories, Labour and the Brexit party.

No 10 would not put a timeframe on publishing the bill, which MPs would expect to happen before the Whitsun recess at the end of next week, but it insisted the vote would happen the week of the 3 June.

“We want to make progress as quickly as possible but we are talking about getting a significant bill which has lots of clauses through both houses of parliament, and we have to allow due time. It’s not possible to give a firm prediction of how long that will take,” the spokesman said.

David Jones, a former Brexit minister and member of the European Research Group of Eurosceptic Tory MPs, told the BBC’s World at One programme a defeat would be the end of the road for this government’s strategy.

“I think actually that No 10 now realise that once this has been introduced, if it is rejected, that is the end of the government’s strategy, and it is really high-stakes politics,” he said.

Jones said May had staked her “personal prestige” on the withdrawal agreement. “She’s had three rebuffs now, and I think it’s very hard to see where she goes after a further rebuff if the bill, when it’s rejected, can’t be reintroduced. If that bill is rejected then it seems to me the whole policy is dead and can’t be pursued any further.”

A spokesman for Jeremy Corbyn confirmed Labour would not support the withdrawal agreement bill unless a cross-party deal was reached. “There is no agreement and we need the government to make further moves,” the spokesman said.

“Without an agreement and real compromise and movement by the government coming out of these talks then we are talking about a withdrawal agreement bill that is based on the same botched Brexit deal that has been rejected three times already by parliament.”

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Talks between officials are expected to continue this week, including May’s chief of staff, Gavin Barwell, but no new meetings have been scheduled between ministers and shadow ministers.

On Wednesday, the leader of the TSSA trade union, Manuel Cortes, urged Labour to pull out of the talks, saying the participation would damage the party at the European elections.

He said: “We are just over a week away from the European elections and it’s now crystal clear that Theresa May has neither the authority nor desire to reach a Brexit deal with our Labour party.

“As a good friend of our party’s leadership, I urge them to please pull the plug on these meaningless talks. I fear we are paying a heavy price on the doorstep for talks which the government has used as a fig leaf for their Brexit failings.”