Update 3.30pm: The UK's Brexit secretary says 'good progress' has been made in talks to try and maintain the Common Travel Area.
David Davis says mechanisms to try and maintain the open border and travel rights are being discussed.
Negotiators between the EU and UK have wrapped up their second phase of talks today, with more due in late August.
Despite claims from Brussels that a border would be inevitable if the UK leaves the customs union, David Davis appeared optimistic.
"Both sides remain committed to the Good Friday Agreement and achieving flexible and imaginative solutions to address the unique circumstances surrounding the border, particularly on the North/South dimension of the agreement.
"These include the mechanisms we have discussed to preserve the common travel area and the rights associated with it."
Earlier: Addressing the impact of Brexit on the relationship between Northern Ireland and the Irish Republic will require "flexible and imaginative solutions", David Davis acknowledged as Brussels said more work needed to be done on the issue.
The European Union’s chief negotiator Michel Barnier said the Government had to clarify how the common travel area would be maintained and also raised concerns about the impact of Brexit on the Good Friday Agreement (GFA).
Brexit Secretary David Davis said the latest talks in Brussels had demonstrated that both sides in the negotiations on the UK’s withdrawal from the EU were committed to the GFA.
Mr Barnier said: "We agree that the important issue of the Good Friday Agreement between Ireland and the UK, in all its dimensions requires more detailed discussions.
"In particular, more work needs to be done to protect North-South cooperation between Ireland and Northern Ireland.
"Today, that cooperation is embedded in the common framework of EU law and EU policies.
"We need to better understand how the UK intends ensuring the continuation of this cooperation after Brexit.
"We also agree that the UK should clarify in the next session how it intends maintaining the common travel area after leaving the EU."
At a joint press conference in Brussels at the conclusion of four days of talks, Mr Davis said: "Both sides remain committed to the Good Friday Agreement and achieving flexible and imaginative solutions to address unique circumstances around the border and particularly on the North-South dimension of the agreement.
"These include the mechanisms we have discussed to preserve the common travel area and the rights associated with it, rights laid down, of course, in a British Act of Parliament as well as, in part at least, in the Amsterdam treaty."
Mr Davis also insisted the Conservative deal with the Democratic Unionist Party at Westminster would not impact on its role in attempts to restore devolved government to Northern Ireland.
He said: "The British Government continues to take incredibly seriously its requirement for impartiality irrespective of any deal of any sort."