Theresa May has been told by Donald Tusk that it is her job to find a solution to the Brexit impasse during what sources have described as an “open and frank” 45-minute phone call in the wake of her demands for a renegotiation.

The European council president warned the prime minister that a precondition for any further talks was a concrete plan from Downing Street that could clearly command the support of parliament.

She in turn insisted to the EU’s most senior official that parliament had highlighted the issue that needed to be addressed in its vote on the so-called Brady amendment on Tuesday evening. But the EU source said May then subsequently failed to offer any proposals during the conversation.

Tusk is understood to have replied that the prime minister could not expect Brussels to come to her rescue with a solution. EU officials and leaders are increasingly concerned that Downing Street is seeking to blame Brussels for their failures.

After the call, which overran as the two leaders grappled over the next steps in the talks, Tusk tweeted: “My message to PM Theresa May: the EU position is clear and consistent. The withdrawal agreement is not open for renegotiation. Yesterday, we found out what the UK doesn’t want. But we still don’t know what the UK does want.”

There are no talks yet planned in Brussels, although the prime minister told Tusk that a face-to-face meeting would be useful in the coming days.

The call came just hours after Michel Barnier, the EU’s chief negotiator, accused May of distancing herself from her own Brexit deal as the EU steadfastly rejected each of the demands the prime minister made in parliament on Tuesday evening over the Irish backstop.

“She took distance from the agreement she herself negotiated and on which we had reached an agreement,” Barnier told MEPs holding a debate in the European parliament. The UK government, he went on, was explicitly supporting an amendment calling for the backstop to be replaced by alternative arrangements that were never defined. “Calmly, I will say, right here and now, we need this backstop as it is.”

Barnier also launched a thinly veiled attack on the former Brexit secretaries, Dominic Raab and David Davis, as he called for a “lucid and realistic” approach from the UK. “When I hear some people who were even part and parcel of the negotiations saying what they’re saying, it’s tough. I find it hard to accept this blame game they’re trying to play against us,” he said.

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On Tuesday night, May celebrated the passing of the amendment tabled by Sir Graham Brady sending her back to Brussels to replace the Irish backstop solution for avoiding a hard border on the island of Ireland.

May said the vote would send a clear message to Brussels about what parliament needed for it to support the withdrawal agreement, citing the need for a time-limit or unilateral exit mechanism from the custom union that is envisaged within it.

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Speaking before Barnier, the European commission president, Jean-Claude Juncker, disputed that claim.

“The withdrawal agreement remains the best and only deal possible,” he said. “The EU said so in November, we said so in December, we said so after the first meaningful vote in the Commons in January. The debate and votes in the Commons yesterday do not change that. The withdrawal agreement will not be renegotiated.”

“We have no incentive or desire to use the safety net but at the same time no safety net can ever truly be safe if it can just be moved at any time,” Juncker said.

“Sometimes from time to time I have the impression that some hope that the 26 other countries will abandon the backstop – and so Ireland – at the last moment, but this is not a game … It goes to the heart of what being a member of the European Union means. Ireland’s border is Europe’s border and it is our union.”

To the prime minister’s wish-list, Barnier also said heads of state and government had already “rejected a limit in time or leaving unilaterally the backstop as that would undermine the very idea of the backstop”.

EU diplomats on Wednesday voiced concerns in a private meeting that the prime minister had failed to achieve a mandate from parliament to secure a specific change, and concluded that the UK would have to ask for Brexit delay to avoid a no deal scenario.

Juncker said the “alternative arrangements” to the backstop backed by the Brexiters and sought by May did not exist, describing it as a “concept not a plan”.

Meanwhile, the European commission announced further no-deal contingency plans affecting students, British researchers, farmers and other recipients of EU funding, citing the increased risk of the UK crashing out of the bloc.

If the UK leaves the EU without a deal on 29 March, the government will be handed a three-week deadline to decide whether to fully honour the £9.7bn budget contribution the UK is due to pay into the EU this year.

If the UK agrees by 18 April to continue payments and accept EU checks and controls, the EU will continue to release funds for British researchers and farmers and others with existing EU contracts.