Theresa May has vowed to bring forward key Brexit legislation for a Commons vote in the first week of June in what could be her last move as prime minister.

After a late-night meeting with Jeremy Corbyn, the prime minister said the cross-party talks with Labour to find a solution to the deadlock at Westminster will continue – despite both sides being downbeat about any resolution being found.

The government made clear a vote on the Withdrawal Agreement Bill (WAB) will be brought forward regardless of the outcome of the discussions, however, in the week beginning 3 June – the same week the US president Donald Trump is to visit the UK on an official state visit.

The announcement came as Ms May’s cabinet set a summer deadline for the UK to finally leave the European Union, after an extended meeting on Tuesday lasting over two hours at Downing Street.

Ministers agreed it was “imperative” to pass legislation ratifying Brexit before parliament rises for the summer break in July, and decided not to terminate the talks with Labour, which are now in their seventh week.

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Following discussions with the Labour leader, a Downing Street spokesman said Ms May told Mr Corbyn the government’s “determination to bring the talks to a conclusion” and deliver on the referendum result.

“We will therefore be bringing forward the Withdrawal Agreement Bill in the week beginning the 3rd June,” they added.

The spokesman also described the ongoing talks – now in their seventh week – as “useful and constructive”, adding further discussions will take place on Wednesday “as we seek the stable majority in parliament that will ensure the safe passage of the withdrawal agreement bill and the UK’s swift exit from the EU“.

But Labour sources told The Independent that Mr Corbyn dismissed any suggestion that Labour would support the prime minister in June without an agreement in the talks in place.

A Labour spokesperson added: “Jeremy Corbyn made clear the need for further movement from the government, including on entrenchment of any commitments.

“In particular he raised doubts about the credibility of government commitments, following statements by Conservative MPs and cabinet ministers seeking to replace the prime minister.”

The “WAB” is not a fourth attempt by Ms May to pass a “meaningful vote” – or MV4 – on her EU Withdrawal Agreement after the three crushing defeats it has already suffered in the House of Commons.

But Brexit legislation demands that MPs approve both the Agreement and the political declaration on future relations agreed with Brussels in November. These could possibly be included as a clause in the bill, rather than a standalone meaningful vote.

The vote on WAB will come during a crucial week in Westminster, with Mr Trump set to spend three days in the UK to coincide with 75th anniversary of D-Day – making a potential defeat for the prime minister even more humiliating.

Voters in Peterborough will also head to the polls in a by-election, after the disowned Labour politician Fiona Onasanya became the first MP to be removed by her constituents last month.

Arch-Eurosceptic Steve Baker, deputy chairman of the Tory backbench European Research Group, gave the WAB little chance of passing.

“If the Brexit Party were demanding we pass this Withdrawal Agreement, it might make sense,” he told The Independent.

“But they aren’t. Quite the reverse. And driving it through over the heads of the DUP appears to eradicate the government’s majority. What are they thinking?”

The remarks from the two leaders came after foreign secretary Jeremy Hunt used a speech shortly after the cabinet meeting to warn that both the Conservative and Labour Party would be “crucified” by voters if they failed to respect the 2016 referendum result.

Referring to the cross-party talks, he told the Wall Street Journal CEO Council conference in London: “There is potential because when you look at the fundamentals it is actually in both parties’ interests to resolve Brexit.

“Because both of us will be crucified by our base if we went into a general election having promised that we would respect the referendum result, not having respected it.

“And I think the lesson at the local elections is that the downside for Labour is as big as the downside for us. I don’t think it’s impossible that there could be a deal there.”

Tom Brake, the Liberal Democrat Brexit spokesman, told The Independent: "There's not a snowball's chance in hell that the WAB will go through.

"Parliament will look very unkindly on an attempt to bring forward a bill to implement a deal that has not been approved by parliament.

"In the absence of a People's Vote or a confirmatory vote, there are no circumstances in which this could command a majority. It is the last roll of the dice for a government which is seeking to give the impression of action on Brexit a few days before it suffers an embarrassing disaster in the European elections."

But there was little enthusiasm in Brussels regarding any developments in Westminster, as a spokesperson for the EU commission said leaders of the 27 member states are on a “Brexit break”.