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 – were 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.

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 – in a move likely to enrage her eurosceptic Tory MPs.

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 blocs 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.

A joint statement by the Commission and the UK government following the first meeting with Mr Juncker on Thursday 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.

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.