Senior Tories have told Theresa May to open talks with Labour as her only hope of salvaging a Brexit deal, after the EU’s outright refusal to renegotiate left her strategy in tatters.

A badly bruised prime minister was urged to stop trying to “go it alone”, accept her proposed agreement is dead and that she needs the help of other parties to push through softer exit terms.

Nicky Morgan, the former education secretary, told The Independent that “cross-party support and proper discussions” were now essential, while Nick Boles, another former minister, said Ms May “must open cross-party discussions”.

The calls came after EU leaders dealt a devastating blow by scrapping written commitments, designed to help Ms May pass her deal through parliament, after disastrous talks failed to achieve a breakthrough.

Brussels’ frustration at the prime minister’s inability to set out clearly what she wanted was laid bare when Jean-Claude Juncker, the European Commission president, branded the UK approach “nebulous”.

At a press conference, Ms May put a brave face on, insisting her Brexit deal remained on track and that talks in the next few days would achieve “further clarification”.

But EU leaders poured cold water on the chances of any changes to satisfy Tory MPs’ demands for a quick exit if the backstop is introduced to prevent the return of Irish border checks.

Emmanuel Macron, the French president, urged Ms May to put her deal to the Commons, saying “this is the only possible agreement”, adding: “Today is clearly for the British parliament to clearly take a stance on the agreement.”

And Leo Varadkar, the Irish premier, dismissed suggestions of another summit in January, revealing he had rejected the prime minister’s call for a legal assurance on the start date for a trade deal.

“That is not possible because it is not in our gift to deliver that, we cannot promise anything that is not in our power to deliver,” he warned.

At home, the Democratic Unionist Party – which is propping up the Tories in power – piled on further pressure by blaming the prime minister personally for the debacle.

Arlene Foster, the party’s leader, said: “This is a difficulty of the prime minister’s own making. The prime minister has promised to get legally binding changes.

“The reaction by the EU is unsurprising. They are doing what they always do. The key question is whether the prime minister will stand up to them or whether she will roll over as has happened previously.”

Jeremy Corbyn said the prime minister had “utterly failed in her attempts to deliver any meaningful changes to her botched deal”, calling for a Commons vote to kill it off without delay

“Rather than ploughing ahead and dangerously running down the clock, the prime minister needs to put her deal to a vote next week so parliament can take back control,” he said.

Nevertheless, Mr Boles said the route to success for Ms May was cross-party talks to “deliver their support for the deal”.

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“The best that can be said for the EU council is this: hopefully this is the last time that Theresa May tries to go it alone,” he tweeted.

“Next week she must open cross-party discussions, and if Corbyn won’t play ball, talk to SNP, Plaid Cymru and backbench Labour MPs.”

Pointing to only 200 supporters for Ms May in this week’s no-confidence vote among Conservative MPs, he added: “What sort of compromise could secure the additional 120 votes she needs? Norway-plus? Second referendum? Permanent customs union? Or some combination?”

Ms Morgan said: “After this week’s events in Westminster and Brussels, the only way the prime minister gets any kind of deal through is with cross-party support and proper discussions to secure that now need to start.”

And George Freeman, Ms May’s former policy chief, tweeted: “The PM should now pivot to a cross-party Brexit Plan B. And stand down after March 29 for the election of a new Conservative leader.”

At her press conference, Ms May admitted confronting Mr Juncker over whether he described her Brexit demands as “nebulous and imprecise”, after footage emerged.

“I had a robust discussion with Jean-Claude Juncker – I think that’s the sort of discussion you’re able to have when you have developed a working relationship and you work well together,” she said.

“And what came out of that was his clarity that actually he’d been talking – when he used that particular phrase – he’d been talking about a general level of debate.”

The prime minister insisted she was “crystal clear” about the assurances that were needed to convince Brexiteers, who are outraged by the Irish backstop plan.

EU leaders released only a bare-bones statement of “reassurances”, deleting pledges from earlier drafts which had said the bloc “stands ready to examine whether any further assurance can be provided”.