In strained talks on Thursday, during which Donald Tusk suggested that Jeremy Corbyn’s plan could help resolve the Brexit crisis, Theresa May and the European commission president, Jean-Claude Juncker, agreed to hold the next face-to-face talks by the end of February.

That move cuts deep into the remaining time, piling pressure on the British parliament to then accept what emerges or face a no-deal scenario.

It is understood that EU officials are looking at offering May a detailed plan of what a potential technological solution to the Irish border might look like, which could be included in the legally non-binding political declaration on the future trade deal.

The blueprint would pinpoint the problem areas and commit to breaching the technical gaps where possible to offer an alternative to the customs union envisaged in the withdrawal agreement’s Irish backstop.

But officials believe it is increasingly likely that any renegotiated deal will only be put to the Commons at the end of March, necessitating even then an extension of the article 50 negotiating period to get legislation through parliament.

On Thursday the German finance commissioner, Günther Hermann Oettinger, suggested the chance of a no-deal Brexit was now as high as 60%.

“If the British side asks for an extension of two or three months and there are reasons for that, I think there’s a good chance that the member states would accept that unanimously,” he said. “But in the eight or 12 weeks there needs to be the possibility of achieving progress and that there must be a withdrawal agreement at the end of that.”

The prime minister’s failure during her meetings in Brussels with EU leaders on Thursday to go beyond her previous suggestions of a time limit and unilateral exit mechanism on the Irish backstop has confirmed fears that the deal’s ratification will go to the wire.

An EU official said it was remarkable that May had not offered “any new concrete proposals” during her 45-minute meeting with Tusk, the European council president, which was described as “OK” and “honest”. Neither did the prime minister have “any clear answers on the timeline”, the official said.

A second official warned of a “vicious circle”, in that the UK appeared unable to make any proposals and the EU was unable to act until it received a request.

EU sources said Tusk had even suggested that Corbyn’s Brexit plan, spelled out in a letter by the Labour leader on Thursday morning, might be a way out of the impasse, but May did not respond to the point.

A senior Downing Street official said it was “welcome that the leader of the opposition is engaging” in Brexit, but declined to go into what May told Tusk. However, it is understood No 10 warned Brussels that the furious reaction of some Labour MP to Corbyn’s letter meant there was uncertainty over the extent of Commons support.

Facing criticism from May, Tusk defended his remarks that “there was a special place in hell” for politicians who campaigned for Brexit without a plan. An EU source said: “He stood by his statement. He remains of the view that while the truth may be more painful, it is always more useful.” The statement was born of “genuine frustration at the mess we are in”, the source said. Tusk later tweeted: “Still no breakthrough in sight. Talks will continue.”

May’s meeting with Juncker was described as “robust”, with the commission president resolutely rebuffing May’s demand for a renegotiation of the withdrawal agreement.

A joint statement from Downing Street and the European commissionannounced new talks on the speed with which the two sides would seek to complete post-Brexit talks on a future trade deal.

May’s critics have voiced fears that slow progress on those trade negotiations would leave Britain trapped in the Irish backstop, under which the UK would stay in a customs union to avoid a hard border on the island of Ireland.

The joint statement said: “President Juncker … 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.”

EU leaders will hold a summit in the Egyptian resort of Sharm el Sheikh on 24 February. The No 10 official played down speculation that this could be the next key Brexit date. “We haven’t confirmed our attendance, and I wouldn’t be thinking in those terms. It’s an EU-Arab summit, primarily.”

During her day in Brussels, May also met the European parliament’s president, Antonio Tajani, and the chamber’s Brexit coordinator, Guy Verhofstadt. “It’s important that Mrs May today assured us that there will be a backstop, that there is no question to remove the backstop,” Verhofstadt told reporters after the meeting.

“If there are problems with this backstop as it is now foreseen in the withdrawal agreement, our proposal is to try to solve the problem in the political declaration because we are open as parliament from day one to upgrade this political declaration, to make this political declaration more binding, more precise. Also on the issue of the backstop explaining and saying very clearly that this is an insurance and not more than that.”

Following her meetings, May said the talks had been “robust but constructive”. She said: “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, and that changes to the backstop, together with the other work we’re doing on workers’ rights and other issues, will deliver a stable majority in parliament.”

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