The British government has strongly insisted it will not allow a border separating Northern Ireland from the rest of the United Kingdom.
But Brussels has stressed the North must abide by EU rules post-Brexit if a hard Border on the island of Ireland is to be avoided on Brexit day, March 29, 2019.
The apparent impasse calls into question whether enough progress can be made on the issue to ensure the negotiations can continue, as the UK wants to pull out of both the single market and customs union.
Brexit Secretary David Davis said EU and UK negotiators have had "frank discussions" on the Border issue. He insisted a conclusion cannot be reached until talks move on to dealing with the future trading relationship between the UK and EU.
He laid down the gauntlet to the EU yesterday, signalling the UK would not tolerate any attempt to create "a new border inside our United Kingdom".
"We respect the European Union's desire to protect the legal order of the single market and the customs union. But that cannot come at the cost of the constitutional and economic integrity of the United Kingdom," Mr Davis said at the end of the latest round of Brexit talks in Brussels.
"We recognise the need for specific solutions for the unique circumstances of Northern Ireland. But let me be clear, this cannot amount to creating a new border inside our United Kingdom."
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His comments came in the aftermath of a leaked European Commission document which stated that the only way to avoid a hard Border is for Northern Ireland to abide by the rules of the single market and customs union.
Taoiseach Leo Varadkar echoed this - but added it did not necessarily mean the UK or Northern Ireland remaining members of either area.
He suggested that a "bespoke" arrangement may be reached as part of the Brexit negotiations and cited the Isle of Man as an example.
"When it comes to the Border between Ireland and Northern Ireland, what we have all agreed to is that there shouldn't be a hard Border, there should be no physical infrastructure along that Border and that there should be no return to the Border of the past," Mr Varadkar said at the British Irish Council Summit in Jersey.
"It is our view, and has been our view for a very long time, that the only way that can be achieved is if the United Kingdom as a whole or Northern Ireland continues to apply the rules of the customs union and the single market.
"That doesn't mean that they have to be members of it, but it would mean continuing to apply the rules of the single market and the customs union. That's the position that we hold and the best way to achieve our common objectives."
Northern Ireland Secretary James Brokenshire indicated that the UK government was ready to be "pragmatic" in seeking a solution to the "unique circumstances" created in the North.
The Border is one of three issues - along with the exit bill and safeguarding expat rights - that Brussels wants commitments on and ideas it deems credible before it decides in December whether to give the green light to move on to talks on future trade relations.
London has consistently said it wants to avoid the return of a hard Border, including no return to any physical infrastructure, but also argues the UK will leave the single market and the customs union. The Government here argues the UK must remain in a form of customs union with the EU if a hard Border is to be avoided.
Meanwhile, the EU has stepped up pressure on Britain to clarify over the next two weeks what financial obligations to the EU it will honour on leaving the bloc in 2019.