Prime Minister Theresa May and Taoiseach Leo Varadkar have spoken on the phone as the Brexit deadlock continues.
A spokesperson for Ms May said they "both agreed about the paramount importance of no hard border or physical infrastructure" at the border between Northern Ireland and the Republic.
They went on: "The Prime Minister said how she recognised the significance of this issue to the people of Northern Ireland and Ireland and how this remained a joint priority for both Governments, and the EU, to resolve.
“The Prime Minister said we are working hard to find a specific solution to the unique circumstances in Northern Ireland that respects the integrity of the UK, the European Union and the Belfast Agreement.
“She added that we are committed to moving together to achieve a positive result on this as well as restoring devolved Government to Northern Ireland. Both leaders looked forward to continuing relations as close neighbours and allies as the negotiations progress.”
Meanwhile a spokesperson for Mr Varadkar said they “took stock of developments since Monday”.
They continued: “The Taoiseach reiterated the firm Irish position regarding the text as outlined by him on Monday. They agreed to speak again over the coming days."
Earlier, the Prime Minister spoke to the DUP leader Arlene Foster.
The party’s Westminster leader Nigel Dodds said he thinks the Irish Government is playing a “dangerous game with their own economy”.
He went on: “While Mr Varadkar may be digging his heels in now, he may find that sections of the Irish business community will be growing increasingly nervous.”
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Meanwhile there have been warnings that no deal on Brexit would be "reckless and dangerous", as "utterly ruthless" dissident republicans would target border officials.
Independent MP Lady Hermon said a hard border would "inevitably" exist between Northern Ireland and the Republic if the UK and EU failed to agree a deal.
She said: "In the event of no deal, we certainly face a hard border and dissident republicans will regard PSNI officers, HMRC officers and UK border officials as legitimate targets.
"I don't want that on my conscience and I don't believe for one moment the Prime Minister wants that either, or the Government."
Earlier, Brexit Secretary David Davis admitted the Government has not carried out an overall assessment of the impact of Britain leaving the European Union.
Sinn Féin MEP Martina Anderson said: "Eighteen months on from the EU referendum, the fact that the Tories still do not have a clear idea of what they are doing should be astonishing, but unfortunately it isn't to those of us who have been involved in this process.
"It is clear the British government have not even done the basics on Brexit so how can they expect to handle complex negotiations, let alone plan for the future."