Enda Kenny has said that he and Theresa May have agreed they do not want "any return to the borders of the past on the island of Ireland".
The Taoiseach met with the new British prime minister at No 10 Downing Street earlier today to discuss the fallout from Britain's vote to leave the EU.
In a statement released after what he described as a "very good" meeting, Mr Kenny said: "We agreed that we would work together to ensure that the benefits of the peace process are preserved in any new arrangements which might emerge regarding the United Kingdom's future relationship with the EU.
"In particular, we both recognised that Ireland is the only EU Member State that shares a land border with the UK. We are in agreement that we don't wish to see any return to the borders of the past on the island of Ireland."
He added they had an opportunity to have a broader discussion on "common issues of concern" such as trade between Britain and Ireland.
Mrs May says both the Irish and British governments are willing to find a mutually beneficial agreement on free movement.
"We benefited from a common travel area between the UK and Republic of Ireland for many years before either country was a member of the EU," she said. "There's a strong will on both sides to preserve it."
Theresa May (left) and Enda Kenny outside Downing Street | Image via @IrelandEmbGB on Twitter
Yesterday, Mrs May visited Belfast in Northern Ireland where she said the UK's relationship with the Republic of Ireland should continue to be in everyone's benefit after leaving the EU.
Mrs May said any talks on leaving the European Union must take into account Northern Ireland's land border with the south.
"We've had a common travel area between the United Kingdom and the Republic of Ireland many years before either country was a member of the European Union - nobody wants to return to the borders of the past" she said.
"What we do want to do is to find a way through this that is going to work (and) deliver a practical solution for everybody."
Roy Greenslade is a commentator with The Guardian and professor at London City University.
He told Newstalk Lunchtime the issue of the Irish border will be a difficult one.
While DUP MP Jeffrey Donaldson has suggested Dublin could become a border control for EU citizens travelling to Northern Ireland.
