Britain's chief Brexit negotiator David Davis has insisted that any deal with the European Union cannot create a new border inside the United Kingdom.
Mr Davis was speaking at the end of the sixth round of talks with his EU counterpart Michel Barnier.
The border on the island of Ireland was a key issue in the latest round of negotiations in Brussels, along with the issues of citizens' rights and Britain's "divorce bill".
Mr Davis' comments come after an internal EU paper has suggested that the avoidance of a hard border on the island of Ireland effectively requires Northern Ireland to remain in the single market and the customs union.
The working paper, seen by RTÉ News, said that in order to avoid a hard border it is essential that there be no divergence of rules on either side of the Irish border.
That scenario would effectively require Northern Ireland to remain in the single market and the customs union.
The working paper is described as an internal document from the EU's Brexit Task Force led by Mr Barnier.
It was circulated to the 27 member states on Wednesday night and presented to British negotiators in Brussels yesterday.
The paper contains the most explicit expression yet by the EU that the only way to avoid a hard border is for the UK to remain in the customs union and the single market, or at least for Northern Ireland to remain within those structures.
It explicitly reflects the Government's thinking that the avoidance of a hard border requires much more than the kinds of technical solutions favoured by the British government.
The paper says that both the EU and UK have already committed to protecting the Good Friday Agreement.
It says achieving this must be done in a way that respects the integrity of the internal market and the customs union, of which Ireland will remain a full member.
This is the key part of the EU Brexit Task Force working paper suggesting NI remain in the customs union & single market pic.twitter.com/wHMG9YFUu3— Tony Connelly (@tconnellyRTE) November 10, 2017
It then says that it seems "essential" for the UK to make sure there is no regulatory divergence from those rules of the internal market and the customs union which are, or may be in the future, necessary for meaningful north-south cooperation, the all-island economy and the protection of the Good Friday Agreement.
In other words, if Britain wants to avoid a hard border, and to protect the Good Friday Agreement, the only real way to do so is that if both parts of the island are in the customs union and the single market.
The Commission document will place more pressure on the Conservative government given its reliance on the DUP for survival.
Northern Ireland remaining inside the customs union and single market, while the rest of the UK was outside, would impose an entirely new structure on the United Kingdom.
Mr Barnier said today's negotiations had made "some progress" but more work was needed on specific issues.
He indicated that Britain had just two weeks to agree to settle its Brexit bill in order to move on to trade talks at a summit in December.
Mr Barnier said the latest round of talks had largely involved "deepening discussions, clarification and technical work".
"Only sufficient progress - that is to say sincere and real progress - on the three main key issues of these negotiations will enable the triggering of the second phase of our negotiation," he said.
Mr Davis said the negotiations had narrowed to "a few outstanding - albeit important - issues".
The UK was "ready and willing" to engage with Brussels "as often and as quickly as needed" ahead of the 14-15 December summit, he said.
"But we need to see flexibility, imagination and willingness to make progress on both sides if these negotiations are to succeed and we are able to realise our new deep and special partnership."
Taoiseach says no demand for any part of UK to stay in Customs Union
Meanwhile, the Taoiseach has said there is no demand from the Irish Government that any part of the United Kingdom should remain in the Customs Union or the Single Market after Brexit.
Leo Varadkar was speaking at a British-Irish Council Summit in Jersey where he was asked about the revelation of the internal EU memo which suggested that Northern Ireland would need to remain in both in order to avoid a hard border.
He said he was "loth to comment in too much detail on papers that have been leaked to the media".
The Taoiseach said there had been agreement that there would no return to the border of the past.
He said: "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 borders of the past.
"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.
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"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."
Mr Varadkar said that what was required was that the language which had been agreed to needed to be turned into a reality.
He said the Irish Government believed the best way to achieve that was to continue to apply the same sort of rules and regulations.
Earlier, Minister for Foreign Affairs Simon Coveney had said that more serious and credible answers on the Irish border issue are needed from the UK in Brexit talks with the European Union before talks can move to consider trade relations.
Speaking on RTÉ’s Morning Ireland, Mr Coveney said Ireland wanted a solution for Northern Ireland and Britain as a whole, to see Britain remain in the customs union, whether through a partnership or an agreement.
He said Fine Gael is holding its party conference in the border area of Cavan this weekend, as it is very concerned about what the future holds in the border area in the context of Brexit.
EU paper likely to prompt resistance from DUP
The DUP has consistently warned it will not accept anything that undermines the union between Northern Ireland and Great Britain.
For the EU to adopt this stance now suggests that it expects the UK to come around to its position if it wants to move to Phase II of the Brexit negotiations, which deals with the future trading relationship.
The EU has said the UK can only qualify to that next phase if there is "sufficient progress" on the financial settlement, EU citizens' rights, and the Irish border.
A deadlock in the negotiations meant there was no breakthrough at the summit of EU leaders on 14 and 15 October.
While the UK's exit bill has been widely seen as the most difficult stumbling block, one EU source has told RTÉ News that the Irish question may turn out to be an even more difficult issue as the deadline of the December EU summit approaches.
On Monday, the Secretary of State for Northern Ireland James Brokenshire said in a speech in Brussels that it would be "impossible" for Northern Ireland to remain in the customs union and single market, while the rest of the UK remained outside.
He did suggest there could be some aspects of agri-food that were handled on an all-island basis, but Irish sources say he did not fully explain how far that would go, or how it would avoid a hard border.
The task force working paper reveals that both the EU and UK have reached an agreement in principle on the Common Travel Area.
"At the last round [of negotiations], agreement was reached in principle on joint principles on the Common Travel Area, which aim to recognise an existing bilateral arrangement between the UK and Ireland," the paper states.
But its articulation of the customs union and single market issues are likely to prompt fierce resistance from the DUP.