Brexit talks are at risk of collapse as a planned EU compromise on the critical question of the Irish border has been branded “unacceptable” by British cabinet ministers.

The Independent has learnt that EU officials believe they have struck upon “the only way” to bring the two sides together on the Irish border in a bid to secure a withdrawal agreement later this year.

But their proposal has already been outright rejected by at least two cabinet ministers, with one going further and branding the EU’s suggestion “bollocks”.

The impasse over the Irish border threatens to bring the talks crashing down with Theresa May’s beleaguered Chequers proposal already lacking support both in Europe and among her own MPs in Westminster.

The Independent now understands that the EU will try to break the deadlock in negotiations by offering the UK a vague political declaration on the future UK-EU relationship in return for a deal on the Irish border.

A well-placed Brussels source said: “This may well prove the only way to respect the EU’s red lines and allow Theresa May to win approval for a deal in the UK parliament.

“The political declaration holds the key to reaching a deal.”

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Since the start of Brexit talks Brussels has insisted the UK sign up to a legally binding “backstop”, which would come into play if no arrangement to avoid a hard border in Northern Ireland is found before Brexit day.

It would see Northern Ireland effectively remain in the EU’s customs union and single market, creating a customs border down the Irish sea – something both Ms May and her DUP partners say is unacceptable.

Under the new proposal, while the EU is still insisting the backstop must be in the legal text of the withdrawal agreement and must see Northern Ireland remaining under customs union and single market rules, they are now indicating there could be “new language” to make it more palatable to Ms May.

Critically, EU officials are suggesting the withdrawal agreement could be accompanied by a non-binding political statement to be agreed with London, that would say the backstop would never be needed because it would be superseded by an eventual trade deal.

As an example of how the statement might be worded, Brussels officials point to a similar one issued by the UK after the cabinet’s Chequers summit in July, which committed the government to accepting a backstop, but ensuring it “would not need to be brought into effect”.

The declaration would then be suitably vague on the broader future relationship to give the prime minister more flexibility over how a future trade deal would evolve during the transition period.

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The briefing appears to corroborate comments made by senior MEP Danuta Hubner, who spoke at an event in Dublin on Thursday saying “political commitments” could be key to breaking the deadlock over Ireland.

But the approach has not gone down well in London, with one cabinet minister who heard about the proposal telling The Independent: “It’s unacceptable. In fact, it’s bollocks.”

A second cabinet minister agreed that the proposal was “simply not going to work”.

A Conservative source told The Independent: “Chequers is in enough difficulty as it is. I can’t see how anything including a vague declaration not to impose the backstop is going to get through parliament.

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“It puts you at the complete mercy of the European Union on whether it would be implemented or not.”

The strength of opposition indicates Ms May could face a further round of cabinet resignations if she were to consider agreeing to such a proposal, with Boris Johnson and David Davis having already quit earlier this year.

A government spokesman said: “We don’t comment on speculation. The proposals we have put forward for our future relationship would allow both sides to meet our commitments to the people of Northern Ireland in full and we are working hard to get a deal on that basis.

“But we are clear the EU backstop proposals are unacceptable.”

In a sign of how little support Ms May’s current Brexit proposals have, Tory Eurosceptics are this week expected to bring forward their own proposals of how the Irish border issue could be solved.

Others in the party are suggesting that the UK should join EFTA, effectively remaining in the single market on a temporary basis, until a longer term trade deal can be struck, while others in the party are pushing for a new referendum.