Theresa May has reached out to Labour for help with a critical issue with her Brexit plan as she pleads with MPs for extra time to secure a breakthrough.

The prime minister struck a conciliatory tone in a letter to Jeremy Corbyn, where she invited Labour to help drum up alternatives to the divisive Irish backstop, which has proved a major stumbling block to getting a deal through the Commons.

But Ms May risked creating a fresh Tory rift by failing to explicitly rule out a customs union with the EU, amid warnings from rank-and-file Conservatives that such a move would amount to an “unforgivable betrayal”.

It comes after the Labour leader took many in Westminster by surprise when he wrote to Ms May laying out his terms for Labour to support a Brexit deal, which included a customs union and guarantees on workers’ rights.

In her response, Ms May said: “When we met, you outlined your concerns about the possible indefinite nature of the backstop, concerns you repeatedly publicly afterwards.

“I hope you will therefore agree with me that seeking alternative arrangements to the current backstop is a necessary step to finding a deal that can command support in parliament.

“One of the things I would like our teams to discuss is the exact nature of those alternative arrangements.”

Ms May has been scrambling to find a way to replace the divisive Irish backstop with an alternative plan, after MPs overwhelmingly rejected her deal last month.

Talks in Brussels and Dublin yielded only stony opposition from EU leaders last week, who have repeatedly said that the Brexit deal cannot be renegotiated.

However Mr Corbyn’s proposals for a softer Brexit were met with interest by the EU, despite the fact that his demand is for a customs union that gives the UK a say on future trade deals the EU might strike – something Brussels appears unlikely to accept.

Ms May did not reject his demands outright, but said: “I am not clear why you believe it would be preferable to seek a say in future EU trade deals rather than the ability to strike our own deals?”

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She also questioned whether the call for “frictionless” trade would mean reneging on Labour’s commitment to end free movement.

​Chief secretary to the Treasury Liz Truss twice refused to rule out quitting if Ms May accepted the demand for a customs union.

She told Sophy Ridge on Sunday: “I appreciate Jeremy Corbyn has come to the table but the reality is what he is proposing does not deliver on what we want as a country.”

Asked if she could stay in office if the government backed a customs union, she said: “I absolutely do not think that should be our policy.

Elsewhere, Tory members warned party chairman Brandon Lewis that “any attempt to tie us to any form of ‘customs arrangement’ would be seen as a shameful sidestep through semantics”.

The open letter said: “This is a red line that must not be breached. The British people have waited three years to enjoy the benefits of life outside of the EU and for a Conservative government to prevent this would be an unforgivable betrayal.

“We urge the party to reverse this avoidable collision course with the British electorate, before it is too late. Ignore us at your peril.”

The development came after the government admitted MPs might not get a chance to vote on the final deal until next month, despite exit day looming on 29 March.

MPs will consider a Valentine’s Day motion on Brexit with a series of amendments, including a Labour push to impose a 26 February deadline for a meaningful vote on the deal.

Police separate clashing Brexit protesters outside Parliament Show all 11 1 /11 Police separate clashing Brexit protesters outside Parliament Police separate clashing Brexit protesters outside Parliament Anti-Brexit protester Steve Bray (left) and a pro-Brexit protester argue as they demonstrate outside the Houses of Parliament Getty Images Police separate clashing Brexit protesters outside Parliament A pro-Brexit protester argues Getty Images Police separate clashing Brexit protesters outside Parliament Police surround the pro-Brexit protester after he confronted Steve Bray, a pro-European protester Getty Images Police separate clashing Brexit protesters outside Parliament A leave supporter is spoken to by a police officer as he argues with a remain supporter, Steve Bray, outside Parliament PA Police separate clashing Brexit protesters outside Parliament MPs in Parliament are to vote on Theresa May's Brexit deal next week after her December vote was called off in the face of a major defeat Getty Images Police separate clashing Brexit protesters outside Parliament A leave supporter is spoken to by a police officer PA Police separate clashing Brexit protesters outside Parliament Police look on as anti-Brexit protesters demonstrate outside the Houses of Parliament. Getty Images Police separate clashing Brexit protesters outside Parliament Police hold back a leave supporter PA Police separate clashing Brexit protesters outside Parliament A police officer speaks with anti-Brexit protester Steve Bray Getty Images Police separate clashing Brexit protesters outside Parliament Police surround a pro-Brexit protester after he confronted a pro-European protester Getty Images Police separate clashing Brexit protesters outside Parliament Conservative MP David Davies, wearing a gopro camera, speaks to anti-Brexit protester Steve Bray Getty Images

In an attempt to take the heat out of Thursday’s clash, the government is instead promising fresh votes by 27 February if no deal has emerged.

Communities secretary James Brokenshire told the Andrew Marr Show: “I think that gives that sense of timetable, clarity and purpose on what we are doing with the EU – taking that work forward and our determination to get a deal – but equally knowing that role that parliament very firmly has.”

Sir Keir Starmer said Labour would stop the prime minister “running down the clock” to exit day next month by drafting an amendment to compel her to offer a vote by the end of the month.

And Labour MPs Peter Kyle and Phil Wilson were poised to launch a new push for a second referendum, writing in The Independent: “It would also be reckless to accept the withdrawal agreement without the approval of the British people.”

Both bids are expected to be among the range of amendments tabled for votes on Valentine’s Day.