Tánaiste and Minister for Foreign Affairs Simon Coveney has said the Government wants to give British Prime Minister Theresa May time and space to manage "difficult political issues".
However, he said he did not want to give the impression that the Government would reverse away from the Brexit border deal that was in place yesterday.
Speaking on his way into Cabinet this morning, Mr Coveney said the Government will work with the British government on presentational issues around the text which had been agreed on the border, but the core meaning must remain as the Irish Government will not be reversing out of an agreement which they felt they had secured yesterday.
He said the Democratic Unionist Party should be listened to but one political party could not decide what is acceptable and what is not for the British and Irish governments and for the EU negotiators, just because the DUP happen to hold the balance of power at Westminster
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Meanwhile, DUP deputy leader Nigel Dodds has accused Ireland of acting in a "reckless and dangerous" way, saying it was putting at risk years of good Anglo-Irish relations over Brexit.
Speaking in Westminster, Mr Dodds said: "The Irish Republic are flexing their muscles and using their current position to try to gain wins for them.
"I don't argue with their desire to advance their interests, but they're doing so in a reckless and dangerous way which is putting at risk years of good Anglo-Irish relations and good cooperation within Northern Ireland."
The Cabinet met throughout the morning and adjourned around midday. It resumed for a second session this afternoon.
Minister for European Affairs Helen McEntee has said she does not accept the DUP's concern over Brexit divergence and said they like everyone else do not want a hard border.
Speaking on RTÉ's Morning Ireland, Ms McEntee said the "text that was agreed was comprehensive" and that it did not just look at the border issue.
She said the British government has been consistently saying that the border issue can be dealt with in Phase 2 of the Brexit negotiations, but for Ireland there has to be an assurance that there is no hard border.
She insisted there would be no changing to the text agreed before moving on to Phase 2.
"I don't think the government would be willing to change the meaning of the text, however, if further clarification was needed then I am sure that is perhaps something that can be looked at.
"That is, however ,something for the EU task force and Theresa May and the UK government to come forward with.
"Our own view is that text that was agreed was sufficient, it gives us clarity moving into phase two that if all else fails there will be no return to a hard border."
McDonald appeals to DUP to 'step back' from their position
Sinn Féin deputy leader, Mary Lou MacDonald said the DUP has been on the wrong side of the Brexit argument from the beginning and appealed for them to "step back" from their position.
"I would appeal to the DUP to understand this is not a case of orange versus green.This is all of us on this island, protecting ourselves and protecting eachother," said Ms McDonald.
A leading DUP member has said the aggressive approach of Taoiseach Leo Varadkar and the Government towards negotiations over Brexit were partly to blame for the collapse of the deal on the border in Brussels yesterday.
Nelson McCausland said the attitude of the Taoiseach and Tánaiste were in stark contrast to to the approach taken by their predecessors Enda Kenny and Charlie Flanagan.
Mr McCausland said both Mr Varadkar and Mrs May should now reflect on the way forward over Brexit and the border.
Meanwhile, former UK foreign secretary Malcolm Rifkind has said it appears there was a press report "which suggested something different to the agreement, and that's why the DUP took fright".
Also speaking on Morning Ireland, Mr Rifkind said a journalist reported yesterday that "Northern Ireland was going to be obliged to be 100% imposing in Northern Ireland, regulations and directives relating to the border that would not exist elsewhere in the United Kingdom as if that was a 100% obligation and that was never what the document said".
Mr Rifkind said he understood it may have been in an earlier draft and that it was removed.
He described Mr Coveney's comments as "very constructive".
Mr Rifkind said there was no difference as he saw it, to what Mr Coveney was saying compared to the position of the DUP or Northern Ireland as a whole.
"Neither side wants there to be a hard border so the question is to clarify the precise wording of what is in the text, hopefully to reassure the DUP that the existing wording is okay," said Mr Rifkind.