Conservative MPs on both sides of the Brexit divide are telling Theresa May that a fresh referendum is becoming “inevitable”, as talks with Labour to break the impasse looked set to fail.

The prime minister faced the warning as she suffered the embarrassment of conceding this month’s European elections will go ahead – and as her own deputy suggested the crisis would drag on until July.

As cross-party talks entered their sixth week, Labour’s Keir Starmer insisted he would force the Tories to end their refusal to contemplate a Final Say referendum as the price of a deal, saying it was “crunch time”.

Significantly, the attempt to put a public vote centre-stage was bolstered from an unlikely source when Daniel Kawczynski, a Tory Brexiteer, predicted it could become the only option “to break the gridlock”.

“If we cannot do this, if this is beyond us, and if we fail, then another referendum is inevitable,” he said, expressing gloom about a deal otherwise.

Ed Vaizey, a pro-EU Conservative, echoed Mr Kawczynski, saying he was “warming to the idea” of a second vote as potentially “the only way to finally have closure on Brexit”.

The comments came as multiple Labour sources told The Independent there could be no deal without a referendum, a demand now being made by around two-thirds of the party’s MPs, it is understood.

Put It To The People march: Best of The Independent’s pictures Show all 24 1 /24 Put It To The People march: Best of The Independent’s pictures Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent Put It To The People march: Best of The Independent’s pictures Put It To The People protest 23 March 2019 Angela Christofilou/The Independent Angela Christofilou/The Independent

Last Friday, Eric Pickles, a former party chairman, become the most prominent Conservative supporter of a referendum, when he gave his surprise backing.

The calls came as talks between Labour and the government broke up without agreement, with both teams preparing to meet again on Wednesday afternoon.

After the three-hour meeting, Rebecca Long-Bailey, the shadow business secretary, said: "Discussions today were very robust and we're having further meetings this week where we hope to make some progress."

She said there was a "willingness on both sides" to work together but there had been no movement towards a customs union, temporary or otherwise.

Ms Long-Bailey would only say another referendum was "one of many options on the table".

Meanwhile, David Lidington, the de-facto deputy prime minister, bowed to the inevitable, by admitting the government had run out of time to stop the European elections, on 23 May, with no ratification of Brexit in sight.

He paved the way for what is expected to be a Tory drubbing by Nigel Farage’s new Brexit Party, by finally admitting: “Legally, they do have to take place.”

Mr Lidington appeared to set a new deadline for ratifying a deal before MPs leave Westminster for their summer break in late July – which would mean MEPs taking their seats at the start of July.

“Ideally, we’d like to be in a situation where those MEPs never actually have to take their seat at European parliament – certainly, to get this done and dusted by the summer recess,” he said.

No 10 played down the significance of the comment, insisting Mr Lidington had merely been “stressing the importance of having this process completed before MPs rise for summer”.

The elections will cost around £150m. To avoid further costs, the withdrawal agreement bill will need to be put before MPs within a week or so – but that will not happen without a Labour agreement not to vote it down.

At the weekend, the acrimony was laid bare when John McDonnell, the shadow chancellor, and other Labour figures accused Ms May of refusing to budge on the party’s central demand for a permanent customs union.

Heading into the talks, Sir Keir, the shadow Brexit secretary, said: “The time has now come to a crunch time, where the government has to decide whether it’s serious about significant changes capable of actually carrying a majority in the House of Commons.”

He pointed significantly to the clamour over a further referendum, adding: “A confirmatory vote will also be on the agenda, and we’re going to discuss that this afternoon.”

No 10 has refused to provide “a checklist” of her continued red lines in the talks, but her own MPs made clear the referendum issue could no longer be dodged.

“If there can be no compromise between the parties, I can actually see then the logic, and other people will be demanding another referendum,” Mr Kawczynski added.

Mr Vaizey, a former minister, said: “I've always been against a second referendum. But I think what is changing my mind is this talk about any agreement reached by parliament being seen as illegitimate.”

Any cross-party deal would pass only narrowly, which means MPs would “spend their whole time telling the public that this isn't Brexit, it’s not the Brexit they voted for – it's illegitimate”.

Earlier, the prime minister met Graham Brady, the head of the 1922 Committee of Tory backbenchers, who was expected to again urge her to set a rapid timetable for her departure.

If she refuses, the 1922 will consider rewriting the rules to allow a fresh vote of no confidence this summer – but Sir Graham made no immediate comment on Ms May’s response.

Asked about the EU elections, the prime minister’s spokesman said she “deeply regrets” the fact the UK would be taking part, admitting: “Many members of the public will feel great frustration at this.”