Theresa May has pledged to give MPs another opportunity to vote on Brexit early next month, with or without Labour’s backing, after Jeremy Corbyn raised concerns about her ability to deliver on a cross-party deal.

The prime minister called a meeting with the Labour leader on Tuesday night to take stock of the Brexit negotiations, as the government sought to inject fresh urgency into the process. The pair held an hour-long meeting alongside the two parties’ chief whips.

A Labour spokesperson said: “[Corbyn] set out the shadow cabinet’s concerns about the prime minister’s ability to deliver on any compromise agreement. In particular, he raised doubts over the credibility of government commitments, following statements by Conservative MPs and cabinet ministers seeking to replace the prime minister.”

But Downing Street insisted that the government was determined to bring forward the key piece of Brexit legislation – the withdrawal agreement bill – in the week beginning 3 June. “It is imperative we do so then if the UK is to leave the EU before the summer parliamentary recess,” a spokesman said.

May’s divided cabinet agreed on Tuesday that the talks with Labour should continue, despite the lack of substantive progress so far, but set a fresh deadline of the summer recess for parliament to pass the bill.

The 3 June date for tabling the legislation suggests the government is effectively setting a deadline for achieving progress in the cross-party talks – although neither side is keen to walk away before the European parliament elections on 23 May.

Labour has repeatedly suggested that May and her colleagues have refused to budge on the central issue of a customs union with the EU.

Fourteen senior Tories, including the leadership contenders Boris Johnson and Dominic Raab, wrote a letter to May on Monday warning that adopting a customs union would alienate the “loyal middle” of the Conservative party.

With the prime minister under intense pressure from her backbenchers to abandon the talks, cabinet ministers held what Downing Street called an “extensive” discussion on Tuesday morning.

“Ministers involved set out details of the compromises which the government was prepared to consider in order to secure an agreement which would allow the UK to leave the EU with a deal as soon as possible,” the prime minister’s spokesman said. David Lidington, who has been leading the Brexit talks for the Tories, gave an overview of progress.

No date has yet been set for parliament’s summer recess, which usually begins in mid-July. MPs are due to go on a Whitsun recess from 23 May – the day of the European elections – to 4 June.

With the Tories apparently heading towards a crushing defeat by Nigel Farage’s Brexit party at the European elections, the prime minister spoke out against the risk of succumbing to “absolutists” on the subject of Brexit, according to one cabinet source.

She reminded colleagues of the need to abide by collective responsibility, in remarks regarded by some as a dig at Jeremy Hunt, who made a wide-ranging speech this week that included a demand for a doubling of defence spending. “She didn’t sound like someone preparing to resign,” said one cabinet source.

The week chosen for the next Brexit vote will also see a state visit by the US president, who will take part in D-day commemorations on Wednesday 5 June.

May hopes that by setting a clear date for the bill, she can fend off calls for an imminent no-confidence vote. She is due to appear before the executive committee of the 1922 Committee on Thursday. Up to 10 of its members plan to demand that she set a date for the end of her premiership or face immediate demands to go.

The party is increasingly struggling to contain widespread anger among its MPs and members at May’s failure to lead the UK through Brexit. Several key committee members, including the chair, Sir Graham Brady, and the treasurer, Sir Geoffrey Clifton-Brown, have said May should draw up a clear schedule for her departure.

The Guardian understands that other executive members, including Sheryll Murray, the South East Cornwall MP, have discussed plans to tell her to stand down immediately if she does not arrive at the meeting with a short, unambiguous timetable for departure. “If she has to be told to her face, so be it. Enough is enough,” said one source.

May’s spokesman reiterated that the prime minister intended to step down once she had completed the first phase of the Brexit talks.

As a crowded field of senior Tories have begun making their pitches for a potential leadership contest, Labour has become increasingly concerned about the impact of May’s fragile authority on the outcome of the Brexit talks.

John McDonnell, the shadow chancellor, speaking at a Wall Street Journal event, said it was becoming hard to see how Labour could “march our troops up the hill” when Johnson and Raab were prepared to overturn any cross-party deal.

Asked whether the talks were going anywhere, he said: “The customs union is absolutely key to us. We are not near what we want.”

McDonnell said Johnson had made clear he was not going to accept a customs union and would overturn any deal agreed in negotiations. “It gives us no security and we expect a response from government to that,” he said.

McDonnell echoed the views of Keir Starmer, the shadow Brexit secretary, that a cross-party deal would attract the support of large numbers of MPs only if a second referendum was attached.

In Brussels, the prime minister’s Brexit negotiator, Olly Robbins, provided EU officials with copies of agreed text that has emerged from the cross-party talks designed to be included in the withdrawal legislation, it is understood.

Brussels was briefed that a meeting of minds had been found in some areas, including parliament’s role in future trade talks and commitments on social and environmental protections demanded by Labour.

But EU sources said there was scant evidence of an imminent breakthrough on the more contentious areas, including a permanent customs union or confirmatory referendum.

Additional reporting by Rowena Mason and Daniel Boffey