The prime minister’s chief Brexit negotiator, Olly Robbins, is travelling to Brussels to discuss changes to the political declaration on the UK’s future relationship with the EU.
The move is intended to meet a key Labour demand in cross-party talks to break the deadlock.
But it comes as senior Conservatives have written to Theresa May warning her not to compromise with Jeremy Corbyn.
Mr Corbyn has also faced demands from his MPs to abandon the talks.
With the two parties seeking to find a compromise over Brexit, it has been a key demand of Labour negotiators that any deal they strike should be reflected in changes to the political declaration.
This declaration was published alongside Mrs May’s withdrawal agreement and sets out the parameters for the future relationship between the UK and the EU, but is not legally binding.
In Brussels, Mr Robbins will explore how quickly changes could be made to the political declaration if the government and Labour can come to an agreement.
But as the cabinet and shadow cabinet take stock of the progress so far in the cross-party talks, pressure is mounting on both sides for negotiations to be abandoned.
Sir Graham Brady, chairman of the Tory backbench 1922 Committee, and 13 former cabinet ministers have written to Mrs May to warn her not to agree a compromise with Labour that includes a customs union with the EU.
Among the former cabinet ministers are Brexiteers Boris Johnson, Dominic Raab and sacked defence secretary Gavin Williamson, as well as Maria Miller and Sir Michael Fallon, who supported Remain in the 2016 referendum.
According to the Times, the letter said that such a deal would lose the support of Conservatives who backed the prime minister in March, and would be unlikely to gain enough Labour votes to pass.
It said: “More fundamentally, you would have lost the loyal middle of the Conservative Party, split our party and with likely nothing positive to show for it. No leader can [bind] his or her successor, so the deal would likely be at best temporary, at worst illusory.”
BBC political correspondent Iain Watson said Labour negotiators were also concerned that any agreement reached with Mrs May would not survive her departure from Downing Street.
And at a meeting of Labour MPs on Monday night, Jeremy Corbyn faced repeated demands to abandon the talks with the prime minister.
MPs fear that they are costing Labour support ahead of the European elections on 23 May.
But speaking after Monday’s discussions with the government, shadow chancellor John McDonnell said they were “constructive as always”.
It comes after shadow Brexit secretary Sir Keir Starmer told the Guardian on Sunday that a cross-party Brexit deal would not get through Parliament unless it was subject to a fresh public vote.
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