DOWNING Street has insisted Britain is sticking by its proposals for keeping the Irish border open after Brexit, amid reports that they have been rejected by Brussels.

A Number 10 spokesman said the UK does not recognise claims that the plans were subjected to a "systematic and forensic annihilation" by EU officials at a meeting this week with Britain's lead negotiator Olly Robbins.

One report of a meeting this week between Robbins and senior EU officials suggested that the Prime Minister's plans for avoiding a hard border with the Republic were subjected to "a systematic and forensic annihilation", and were comprehensively rejected.

The Daily Telegraph quoted unnamed EU diplomatic sources as saying that the Brussels officials delivered "a detailed and forensic rebuttal", making clear that "none of the UK customs options will work – none of them".

The report came as Britain's former ambassador to the EU, Sir Ivan Rogers, said that UK hopes of finding a technological solution to the border issue were regarded as "a fantasy island unicorn model" in European capitals.

A UK Government spokesman insisted that Britain was "continuing an intensive work programme to engage" on all the scenarios set out in the Joint Report agreed in December by May and European Commission president Jean-Claude Juncker.

The agreement included three options for the Irish border, with the British backing a close economic relationship which would make customs checks unnecessary or technological solutions to render them near-frictionless.

But a version published by the EU in February contained only the third "backstop" option, effectively drawing a customs border down the Irish Sea, which a furious May said "no British Prime Minister could ever agree".

The report puts pressure on May ahead over a vote in the Commons next week on keeping the UK in the European Customs Union, just days after she was defeated on the issue in the Lords.

If the UK position is rejected by Brussels, the Government could be faced with a choice between remaining in the Customs Union or accepting a hard border in Ireland.

European Council president Donald Tusk has warned that there will be "no withdrawal agreement and no transition" without a solution on Ireland.

Robbins was also warned that Brussels needs "full compliance" with EU rules on goods and agricultural products in the whole of the UK – not just Northern Ireland – if customs barriers are to be avoided.

A Government spokesman said: "We have been clear that we will protect Northern Ireland's place in the UK internal market.

"That commitment was set out in December's Joint Report which also includes our guarantee of avoiding a hard border between Northern Ireland and Ireland.

"As the PM's letter to Donald Tusk said, we have made our position on aspects of the draft Commission Protocol clear.

"We have agreed that the areas covered in the draft must reflect those that meet our shared commitments.

"And we are continuing an intensive work programme to engage on all the scenarios set out in the Joint Report."

Responding to the reports, former Treasury permanent secretary Lord Macpherson tweeted: "EU's position on Irish border so predictable. UK sold pass by conceding 'backstop' in December, inviting EU to hold us to it. #badbusiness".

Speaking at the Policy Exchange think tank in London, Rogers said that the economic solution was not regarded as "a runner" on the other side of the Channel, because of May's insistence on leaving the single market and customs union.

And he added: "The Brits are therefore focused above all on Option B – the technological solution. That, candidly, from everything I've heard from various places, is still viewed as a bit of a fantasy island unicorn model.

"The Irish and Brussels in particular – but I think backed, as far as I can see, by Berlin and Paris – have said the only solution to this is the so-called backstop Option C, which is what the Commission put in print and got the toxic reaction both from the DUP and the Prime Minister."

Meanwhile, there have been reports of frustration within the Cabinet over delays in drawing up the Government's plans for immigration after Brexit.

Home Secretary Amber Rudd told MPs last month that the immigration bill would not be introduced until early next year.

But The Times quoted one unnamed source as saying Rudd seemed to think she could "take as long as she likes", and said an unnamed "ally" of Work and Pensions Secretary Esther McVey had said: "We're eager to get on with it".