British Prime Minister Theresa May's hopes of a Brexit breakthrough are hanging in the balance after the EU appeared to rebuff her latest appeal for help in getting her withdrawal deal through Parliament.

The EU's chief negotiator Michel Barnier said they would be "working intensively" in the coming days to resolve the impasse over the Northern Ireland backstop ahead of Tuesday's crunch vote by MPs.

However, his suggestion the UK could unilaterally pull out of the planned single customs territory if it chose was dismissed by ministers as an attempt to "rerun old arguments".

The DUP, which props up the British government at Westminster, said Northern Ireland would still be in the backstop, effectively drawing a "border in the Irish Sea" with the rest of the UK, something Mrs May has repeatedly ruled out.

There had been speculation Mrs May could fly out to Brussels early on Monday to clinch a deal if an agreement on the backstop - intended to prevent the return of a hard border on the island of Ireland - was within grasp.

She had been seeking legally binding assurances that the UK could not be tied indefinitely to the EU through the backstop, in an attempt to win round MPs who inflicted a massive defeat on the Withdrawal Agreement in a previous vote in January.

However, in a sign ministers are preparing to go back to the Commons on Tuesday without fresh concessions, Chancellor Philip Hammond said such fears about the backstop were overblown.

In an interview with the Financial Times, he urged Tory MPs to back the Withdrawal Agreement anyway, saying it would enable him to start spending the "insurance fund" built up in case of a no-deal break.

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"There is nobody in the EU I've ever come across who thinks the UK could be held in perpetuity in an arrangement that was detrimental to its interests against its will. Who is going to enforce such an arrangement on this?" he said.

Mr Barnier's latest intervention came after Mrs May delivered a speech yesterday imploring EU leaders for "one last push" to help her get her deal over the line in the Commons.

It was unclear whether his comments, in a series of tweets, were intended as a new offer or simply a restatement of the EU's position after his talks on Tuesday with Attorney General Geoffrey Cox and Brexit Secretary Stephen Barclay failed to reach agreement.

He said the EU was committed "to give UK the option to exit the Single Customs Territory unilaterally, while the other elements of the backstop must be maintained to avoid a hard border".

The EU was also ready to give "legal force" to reassurances given to Mrs May concerning the operation of the backstop in January in a joint letter from European Council president Donald Tusk and European Commission president Jean-Claude Juncker.

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Mr Barclay, however, suggested the latest proposals from Brussels were simply a return to an earlier version of the backstop which had already been rejected by the UK.

"With a very real deadline looming, now is not the time to rerun old arguments," he said.

"The UK has put forward clear new proposals. We now need to agree a balanced solution that can work for both sides."

DUP deputy leader Nigel Dodds said Mr Barnier's offer would annex Northern Ireland from the rest of the UK- something Mrs May has repeatedly said that no British prime minister could agree to.

"This is neither a realistic nor sensible proposal from Michel Barnier. It disrespects the constitutional and economic integrity of the United Kingdom," he said.

"This is an attempt to get ahead of a possible blame game and appear positive when in reality it is going backwards to something rejected a year ago." 

Tory demands would lead to "absolute chaos" - McDonald

Sinn Féin leader Mary Lou McDonald says the Irish Government needs to hold firm in the Brexit negotiations to avoid chaos on this island.

"I think the British government assumed that they could apply pressure tactics and that either Europe or Dublin would give way to the Tory demands would mean absolute chaos on this island."

She said "the backstop is the bare minimum that is required", so that people in Northern Ireland are not left behind.

"If Britain wishes to Brexit who are we to stop them, that's their right, but they aren't taking the North of Ireland with them, and they certainly aren't going to wreck the Irish economy and upend our peace process."

Mary Lou McDonald is to visit the United States next week, where she will be asking for support for the Good Friday Agreement, so that Britain is reminded it has international obligations under law.

Asked whether the Secretary of State for Northern Ireland Karen Bradley should resign over comments made this week, Mary Lou McDonald said "it is a resigning matter".

"I think she should go but I'm also saying the problem is much bigger than Karen Bradley. This is isn't down to one individual."

"Karen Bradley played her hand and said out loud what the British position is, and the position is to withhold information."

"The British system refuses to come forward with the truth, refuses to acknowledge the level of collusion between the British government and its loyalist proxies, the death squads in the North."