Theresa May’s team are in behind-closed-door talks in a bid to secure the support of up to 20 Labour MPs for the prime minister’s troubled Brexit deal.

Ministers are said to be negotiating with the MPs from Leave-backing seats, as pressure intensifies on Ms May to secure backing for her twice-defeated plan.

With just four days to go until the vote, the Labour Brexiteers have demanded that parliament’s right to shape Britain’s future relations with the EU be cemented into law.

Attempts to woo the support of Jeremy Corbyn’s MPs have so far largely failed, but if efforts succeed then the Labour backing could neutralise an expected rebellion of hardcore Tory Eurosceptic MPs.

It comes as senior ministers also held “significant” talks with the Northern Irish DUP in a bid to lock in their support for the big vote on Tuesday, while more Tory MPs came out in support of Ms May’s deal as the clock ticks down to Brexit day on 29 March.

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Up to now Ms May’s efforts to win over Labour Brexiteers to her deal have fallen short, as her £1.6m fund for Leave-voting towns in England was met with mockery, while a separate package to protect workers’ rights persuaded neither MPs nor trade unions.

A handful of Labour MPs are thought to want to be more certain that the deal will pass before publicly backing it, but a second key problem is trust, with others unwilling to accept assurances that a new Tory PM will not ditch promises over employment rights a few months down the line.

One Labour MP involved told The Independent: “If we get the assurances we want, then up to 20 Labour MPs could back her when the vote comes back.

“And it means that even if she is ousted by her own MPs, then we have those assurances laid down in law.”

Labour Leavers want ministers to give legal force to commitments on employment rights and standards that currently sitting in the Brexit political declaration, and also to provide fixed dates on which parliament will be consulted on them.

Another Labour MP said: “Parliament should have an opportunity to be able to express views on all of the different aspects – that was said, it should not be a problem – but it never materialised in writing.

“What colleagues were looking for was something much more definitive from the prime minister.”

The MP added: “It could be along those lines because that was one aspect that was supposed to be attractive to a number of colleagues.”

MPs have overwhelmingly rejected the prime minister’s deal twice – first by 230 votes and then again by 149 on Tuesday – but the PM is hoping one more push could take it over the line.

Ms May had called Labour MPs personally ahead of the second vote on her deal on Tuesday, but still suffered a historic defeat, with only three backing the deal.

But the potential to win Labour support is clear. For example, 18 of Mr Corbyn’s MPs defied his will to ensure a second referendum was not backed by parliament this week, five of whom were frontbenchers who later resigned.

Mr Corbyn had ordered his MPs to abstain but shadow ministers Yvonne Fovargue, Emma Lewell-Buck and Justin Madders, plus party whip Stephanie Peacock, all voted against the motion and then quit.

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Stoke-on-Trent North MP Ruth Smeeth, a parliamentary aide to deputy leader Tom Watson, also resigned.

Ms Lewell-Buck told her local paper: “If it comes down to either voting for Theresa May’s deal, having no deal or not leaving then I’ll end up voting for her deal, because I’ll be backed into a corner.”

Six Labour MPs also helped Ms May block a bid from a cross-party group to strip the government of control of the Brexit process, something which had been expected to pass.

On Friday, it emerged that chancellor Philip Hammond is leading government talks with the DUP in a bid to win their support for the deal, prompting speculation there could be financial incentives for the party to back the plan.

When the DUP agreed to prop up Ms May’s administration in the Commons after the 2017 election, the party negotiated more than £1.5bn in extra spending for Northern Ireland over 2018-19.

Money in the agreement, and potentially the support of the party’s 10 MPs for Ms May, is set to end in June, creating the opportunity for its leader Arlene Foster to negotiate new terms, with the government’s main ask likely to be safe passage of her Brexit deal.

Following top level discussions on Friday, which also involved Cabinet Office minister David Lidington, environment secretary Michael Gove and chief whip Julian Smith, DUP deputy leader Nigel Dodds said there had been “a constructive dialogue”.

Asked if extra cash for Northern Ireland had been discussed with Philip Hammond, Mr Dodds said: “The chancellor of the exchequer is obviously a key member of the government, but he is also responsible for HMRC and the whole issue of their involvement in customs and other regulatory issues is a key concern for us.”

He added: “We are not discussing cash in these discussions.”

In a further sign of the weakening opposition from Eurosceptic Tories, Esther McVey, who quit the cabinet over Ms May’s deal, indicated she would now back it.