Theresa May has spoken to Jean-Claude Juncker as part of her effort to obtain further written reassurances that the Irish border backstop in her Brexit deal would never come into force.

The phone call between the prime minister and the European commission president on Friday was described by Brussels as “friendly” but Downing Street refused to provide any further details, as May struggles to break the Brexit impasse.

She is still hoping to obtain additional clarifications over the backstop before MPs vote on the deal in the week of 14 January, although it is not clear what May can get that would satisfy Tory rebels and the Democratic Unionist party.

Quick Guide Brexit and backstops: an explainer Show A backstop is required to ensure there is no hard border in Ireland if a comprehensive free trade deal cannot be signed before the end of 2020. Theresa May has proposed to the EU that the whole of the UK would remain in the customs union after Brexit, but Brussels has said it needs more time to evaluate the proposal. As a result, the EU insists on having its own backstop - the backstop to the backstop - which would mean Northern Ireland would remain in the single market and customs union in the absence of a free trade deal, prompting fierce objections from Conservative hard Brexiters and the DUP, which props up her government. That prompted May to propose a country-wide alternative in which the whole of the UK would remain in parts of the customs union after Brexit. “The EU still requires a ‘backstop to the backstop’ – effectively an insurance policy for the insurance policy. And they want this to be the Northern Ireland-only solution that they had previously proposed,” May told MPs. Raising the stakes, the prime minister said the EU’s insistence amounted to a threat to the constitution of the UK: “We have been clear that we cannot agree to anything that threatens the integrity of our United Kingdom,” she added.

Expectations are being kept deliberately low, and May is likely to begin the political year talking about the long-term future of the NHS on BBC One’s The Andrew Marr Show on Sunday.

May has recently spoken to the president of the European council, Donald Tusk, and the German chancellor, Angela Merkel. The discussions were initially confirmed by the offices of Tusk and Merkel. No 10 will not otherwise say who the prime minister has spoken to, unless the counterpart does so first.

A European commission spokeswoman described the call between Juncker and May as friendly” and said they “agreed to stay in touch next week” when Westminster returns from its Christmas break and the debate on the meaningful vote will restart.

The Commons will discuss May’s Brexit deal on Wednesday after prime minister’s questions, continuing on Thursday and possibly Friday. The final vote, pulled last month after the prime minister admitted she would lose by a “significant margin”, is due to follow in the middle of the week following.

Time is narrowing for May’s deal to be approved before 29 March, because once MPs agree in principle, ratification legislation will have to be voted through by parliament, a process that normally takes months.

Failure to agree would mean the UK would crash out without a deal, and it would be difficult for MPs to force the government into proposing an alternative. But ministers would need to pass dozens of items of emergency legislation. Labour has said it would seek to introduce amendments that would make no deal difficult to implement.

Downing Street had hoped Brexit passions would have cooled over the Christmas break as warnings mount about the potential impact of no deal, but the early signs are that rebel positions remain firm and the vote is expected to be lost.

It is believed there are 30-40 Tory MPs deemed certain to vote against May’s deal, including former cabinet ministers and veteran anti-EU campaigners, although that would be a significant reduction from the 100 who publicly declared their opposition in December.

The DUP, whose 10 MPs help keep May in power, has been repeating its objections to the proposed Brexit arrangement over the past 24 hours, despite continuing talks with No 10 to try to resolve the impasse.

Sammy Wilson, the DUP’s Brexit spokesman, said his party opposed the backstop because if it came into force, Northern Ireland “would have to treat the rest of the United Kingdom as a third country” and would not participate in any trade deals that the UK may enter into in the future.

Downing Street has floated the idea of staging a second parliamentary vote almost immediately after the first in an attempt to pressure MPs to change their minds, in a crisis atmosphere of rising public concern and jittery financial markets.

The argument is helped by May’s position being more secure than it was in December after rebel MPs failed to unseat her. Under Conservative party rules, she cannot be challenged again for nearly a year.

Labour has repeatedly called for May to step aside if her deal is voted down, but she could only be forced out if she were defeated in a Commons vote of no confidence. Rebel Tories have said they would not vote with Labour in such a situation.

EU sources said on Friday they did not know what May was aiming for with her round of holiday telephone diplomacy. “That is still not clear,” said one diplomat, adding that the “surgical removal” of the backstop demanded by some Eurosceptics was “not remotely possible”.

A call by the foreign secretary, Jeremy Hunt, for the EU to define the temporary nature of the backstop is not seen as realistic. “Define it how?” the diplomat said. “It can’t be a unilateral departure from the backstop and it can’t be time-limited. All of this tends to run around in circles.”

Meanwhile, the Brexit secretary, Stephen Barclay, told a group of European newspapers that EU leaders could not hope to push May into organising a second referendum because it would inflame political tensions.

“The extent to which Britain is divided currently is small compared with the tensions a second vote would cause. It would continue to split our nation,” Barclay said, in remarks first reported by Germany’s Die Welt.

The minister said it would not be possible to hold a referendum before the EU parliamentary elections, due in May, and if Britons had to vote for MEPs again, he predicted there would be “huge democratic damage, because the citizens voted to quit”.

Barclay even signalled that holding European elections in such a context could lead to a surge in support for Ukip or other anti-EU populists. “Even our European colleagues cannot be interested in it because that would trigger a very populist reaction,” Barclay said.

A poll of 1,215 Conservative party members found 57% would support a no-deal Brexit if there were a three-way referendum in which May’s Brexit deal and remaining in the EU were the other two options. In comparison, 25% of voters would back no deal.