Theresa May has declared she “cannot agree to anything that threatens the integrity of our United Kingdom” as she told sceptical MPs the EU’s insistence on retaining the Northern Ireland backstop was unacceptable.

The prime minister said while she had proposed the UK would remain in a countrywide “customs solution” as an alternative, the EU had told her there was not enough time to work out the detail in the next few weeks.

Brussels negotiators had demanded the existing Northern Irish backstop – in which part of the UK would remain in the customs union after Brexit – be retained even though it had been rejected by the government.

“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 on Monday.

“We have been clear that we cannot agree to anything that threatens the integrity of our United Kingdom. And I am sure the whole house shares the government’s view on this.”

The prime minister appeared in front of MPs for an hour and 40 minutes, in a hastily scheduled statement in which she was peppered with questions on the progress of negotiations from sceptical members on both sides of the house.

May heads to Brussels on Wednesday for a critical two-day European summit, where EU leaders have to decide if sufficient progress is being made in the Brexit talks to hold a concluding summit in November, and whether to abandon their proposed Northern Ireland backstop in favour of her alternative.

The prime minister will speak to the French president, Emmanuel Macron, on Monday night, as she tries to lobby EU leaders to change their minds about the backstop. No 10 said she had also spoken to the German chancellor, Angela Merkel, and the Dutch prime minister, Mark Rutte, in recent days.

May said she also needed to ensure her proposed backstop, known as the “temporary customs arrangement”, was time-limited: “I need to be able to look the British people in the eye and say this backstop is a temporary solution.”

The issue of how to time-limit the backstop has caused concern in the Conservative party, with some in the cabinet calling for a firm date, while others have indicated the UK could propose a mechanism by which the backstop would disappear using other criteria.

“People are rightly concerned that what is only meant to be temporary could become a permanent limbo – with no new relationship between the UK and the EU ever agreed,” May told the Commons.

She told MPs talks were entering the final stages. “This is the time for cool, calm heads to prevail and it’s the time for a clear-eyed focus on the few remaining but critical issues that are still to be agreed,” she said.

The prime minister insisted “real progress” had been made on the withdrawal agreement and the political declaration on future relations, and there had been a “great deal of inaccurate speculation” about how the talks were progressing.

“I continue to believe that a negotiated deal is the best outcome for the UK and for the European Union. I continue to believe that such a deal is achievable. And that is the spirit in which I will continue to work with our European partners,” she said.

Jeremy Corbyn urged May to “put the country before her party” and stand up to the “reckless voices” on the Tory benches.

Amid heated scenes in the Commons, the Labour leader said: “For too long, this country has been held hostage to those in her party who want to drive through a race-to-the-bottom Brexit deal that lowers rights and standards, sells off our national assets to the lowest bidder.

“It is clear that the prime minister’s failure to stand up to the warring factions of her own side have led to this impasse.”