Jean-Claude Juncker has told Theresa May in a private phone call that shifting her red lines in favour of a permanent customs union is the price she will need to pay for the EU revising the Irish backstop.

Without a major shift in the prime minister’s position, the European commission president told May that the current terms of the withdrawal agreement were non-negotiable.

Details of the call, contained in a leaked diplomatic note, emerged as Juncker’s deputy, Frans Timmermans, said there had been no weakening of the resolve in Brussels in support of Ireland, and accused the Tory Brexiters of a “cavalier” approach to peace.

“Let me be extremely clear: there is no way I could live in a situation where we throw Ireland under the bus,” Timmermans said. “As far as the European commission is concerned, the backstop is an essential element for showing to Ireland and to the rest of Europe that we are in this together.”

On Tuesday, the Commons will vote on a series of amendments that might variously force the prime minister to delay Brexit or go back to Brussels to demand the ditching of the Irish backstop or a time limit on its enforcement.

Critics of May’s deal believe that the backstop, an “all-weather” solution for avoiding a hard border on the island of Ireland, could trap the UK in an indefinite customs union, limiting the country’s ability to pursue an independent trade policy. May’s deal was rejected this month by a historic 230 votes.

At a confidential meeting of the ambassadors in Brussels last week, EU officials said there had been no back-channel talks in recent days with Downing Street nor talks between May and other heads of state and government.

The European council’s Danish secretary general, Jeppe Tranholm-Mikkelsen, told the ambassadors he believed the impasse could continue into February.

“Next week there is a vote,” he said. “There have been no talks in the last week. I do not expect any clarity or any position in the course of next week. This will take time. In the meantime, pressure will increase. On our end we will need to find ways to deal with that.”

A commission official said: “Yes, difficult, and President Juncker told Mrs May that the backstop was non-negotiable and only if May changed red lines then we can move on the backstop.”

It is understood that one solution discussed by senior officials would be for the backstop to be downgraded to be Northern Ireland specific, as was originally proposed by the EU, should there be a commitment from the British government to negotiate a permanent customs union with the whole of the UK.

But Timmermans said attempts to rip out the backstop from the withdrawal agreement or put a time limit on it were doomed to fail. Delaying Brexit, he added, did not solve the problem of finding an agreement that could avoid a no-deal scenario.

“The problem is that the House of Commons can say they don’t want a no-deal Brexit but if they don’t say what they want there will be a no-deal Brexit on the 29th,” Timmermans said. “The thing with Europe is that if you know our history so many things have happened in our history that nobody wanted that I think we need to be well prepared for a no-deal Brexit.”

“Come together around an idea, and we will listen,” Timmermans told MPs. “Mind you, a backstop is called a backstop because it doesn’t have a time limit. If it has a time limit it is no longer a backstop, so that backstop for the European Union is very important. And there can be no uncertainty about that.”

Frans Timmermans, first vice-president of European Commission, said attempts to rip out the backstop or put a time limit on it were doomed to fail.

Photograph: Jacek Bednarczyk/EPA

Asked whether May should respond to Juncker’s advice and move to support a permanent customs union, Timmermans said: “Well, that is an internal debate in the UK. I am just looking at the number 230. Yeah.

“I don’t know what [Juncker] has been saying privately, I am not privy to that. I am just observing the debate and she will have to command a majority in the House if she wants something done.”

Brexiters were given hope in recent days by the EU’s prevarication over whether a hard border would be enforced in the event of a no-deal Brexit, and by the suggestion from Poland’s foreign minister, Jacek Czaputowicz, that a five-year time limit could solve the problem of the backstop.

“I am not quite sure minister Czaputowicz was speaking on behalf of the Polish government,” Timmermans said. “I have not seen the Polish government repeat that.”

The commission’s first vice-president added he would expect the EU’s leaders to have an open mind towards an extension of the negotiating period beyond 29 March.

“One of the things that irks me most in all of this is this image that has been created of this vindictive unfriendly European union,” he said.

“I think, from where I stand, we have been bending over backwards. Mrs May’s red lines are Mrs May’s red lines ... I also hear people in the UK always saying, ‘oh God, they have been treating us so badly.’ Frankly, I really don’t understand how you can say that.”