The European parliament’s Brexit chief has welcomed Jeremy Corbyn’s new EU exit plan and endorsed calls for a cross-party conensus in Westminster to prevent a no deal.

Theresa May travelled to Brussels on Thursday for meetings with representatives from the European Commission, European Council and European parliament after her MPs told her they could not support the deal she had struck unless a controversial “backstop” cause was removed.

Though the European Commission and UK government agreed to “find a way through” and restart meetings, all the EU chiefs Ms May spoke to were emphatic that the withdrawal agreement – and thus the backstop – was not up for negotiation.

But Brussels is putting increasing pressure on the prime minister to think again about her red lines for the future relationship between the UK and EU – such as on a customs union, or alignment to the single market.

EU officials hope that if Ms May moves her position closer to that of Labour there could be a majority to pass the deal. She is said to have told leaders behind closed doors that she accepts the backstop cannot be scrapped – in a move likely to enrage her Eurosceptic Tory MPs.

Speaking after a meeting with the prime minister in Brussels, Brexit coordinator Guy Verhofstadt said “we welcome” the letter the Labour leader had sent to the prime minister laying out his party’s alternative proposals.

“We have reiterated that we cannot have an agreement with uncertainty in the UK based on majorities of six, seven, eight, nine votes in the House of Commons,” Mr Verhofstadt said in a joint press conference with the parliament’s president Antonio Tajani.

“That cross-party cooperation is the way forward and I think I can say that we welcome the letter that Jeremy Corbyn has written to Mrs May to offer such a cross-party exit, I should say, to the Brexit.

“It’s important now that this leads to a position in the UK that has the broadest possible majority so that we can conclude this negotiation.”

He also confirmed that Ms May had in the meeting not sought to scrap the backstop and conceded it was necessary, adding: “Mrs May today in the meeting assured us that there will be a backstop. What she said already in Belfast: there’s no question to remove the backstop, because that is absolutely necessary for securing and safeguarding the Good Friday Agreement, internal market, and peace process.”

The prime minister is understood to have received a similar message from Donald Tusk, the European Council president who represents the collective views of member states.

Sources with knowledge of the meeting between Mr Tusk and the prime minister said the president told Ms May that “the Corbyn plan could be a promising way out of the impasse”, according to Buzzfeed News.

EU officials who examined the Labour plan believe it to be broadly in line with EU negotiating guidelines, the website says.

Labour last night called for a “permanent and comprehensive UK-wide customs union”, close alignment with the single market underpinned by “shared institutions”, protections on workers rights, and UK participation in some EU agencies. The policy moves the party closer to a so-called “Norway+” Brexit that would keep the UK in the bloc’s orbit, though technically outside.

In a joint statement the British government and European Commission said Ms May had had a “robust but constructive” meeting with president Jean-Claude Juncker, and that the pair would meet again before the end of the month.

The two negotiating teams have been formally stood down since the withdrawal agreement was agreed between the two sides last year, with the EU saying it did not anticipate any further meetings. But with MPs in Westminster blocking the ratification of the agreement, officials will begin meeting again.

The joint statement said the gathering was “held in a spirit of working together to achieve the UK’s orderly withdrawal from the EU”.

Theresa May is said to have explained the situation in Westminster to Mr Juncker and raised “various options for dealing with these concerns”.

But Mr Juncker responded by underlining that the EU27 would not reopen the withdrawal agreement – describing it as ”a carefully balanced compromise between the European Union and the UK, in which both sides have made significant concessions to arrive at a deal”.

He however “expressed his openness to add wording to the political declaration agreed by the EU27 and the UK in order to be more ambitious in terms of content and speed when it comes to the future relationship between the European Union and the UK”, according to the read-out of the meeting.

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The statement concluded: “The discussion was robust but constructive. Despite the challenges, the two leaders agreed that their teams should hold talks as to whether a way through can be found that would gain the broadest possible support in the UK parliament and respect the guidelines agreed by the European Council. The prime minister and the president will meet again before the end of February to take stock of these discussions.”

Ms May’s meeting Mr Tusk comes a day after he caused a row by warning that there was a “special place in hell” for those who promoted Brexit without even a basic plan for how to actually enact it. Speaking after her meeting she said: “I’ve raised with President Tusk the language that he used yesterday which was not helpful and caused widespread dismay in the UK. The point I made to him is that we should both be working to ensure we should deliver a close relationship between the UK and EU in the future and that’s what he should be focusing on.”

Speaking on her way back to London, she added: “I’ve had a good series of meetings today. We’ve had robust discussions but they’ve been constructive.

“What I’ve set out is our clear position: that we must secure legally binding changes to the withdrawal agreement to deal with the concerns that parliament has over the backstop. Taking that: changes with the backstop, together with the other work we’re doing on workers’ rights and other issues, will deliver a stable majority in parliament. That’s what I’ve continued to push for.”