The Taoiseach has said there can be no further concessions from the EU to win the support of the DUP for the Brexit agreement.
Leo Varadkar said that "the deal is done" and that this was the second agreement the EU has made with the UK.
He said he was disappointed that there was no a vote on the Brexit deal in the House of Commons yesterday.
Speaking to RTÉ News as he arrived at Parnell Park for the Dublin hurling final, he said that while an extension would be preferable to no deal, that would require unanimity among the EU27.
He said to get that the UK would need to outline the reasons for an extension, whether that would be for an election or a second referendum.
Meanwhile, MEP and Vice President of the European Parliament Mairead McGuinness has said there was no rush or urgency to decide on another exit date for the UK from the EU.
Speaking to the BBC, she said there was time even up until 31 October for Europe to say if there is a need for a technical extension.
She said it was clear British Prime Minister Boris Johnson does not want an extension and said the EU was remaining calm at this point.
Ms McGuinness said: "Nobody could have anticipated the twists and turns, the votes, the amendments, the court cases and all that has happened.
"No one would have anticipated this. That is why we do have to be calm because if we react to yesterday's events with any degree of urgency or frenzy then tomorrow might give a different outcome.
"And we have to be balanced about it."
She did, however, also say there is a desire to move from this stage, the divorce settlement, which is taking much longer than both sides had anticipated, so they can move on to the future relationship.
Elsewhere, Minister for Social Protection Regina Doherty has said Mr Johnson's letter to the EU, seeking a Brexit deadline extension, could be the subject of a legal challenge.
Speaking on RTÉ's Morning Ireland, Ms Doherty said she believed that lawyers in the UK would be examining the text and "probably will challenge it this week".
The minister said the Irish Government was "disappointed" that the House of Commons was not in a position to pass the latest EU-UK Brexit deal yesterday, adding that it had been hoping for a "clear result".
She said that the Government "really, really hopes" that there could be an agreement signed-off by MPs later this week.
Minister for Health Simon Harris said the EU needs clarity from the British government, and Mr Johnson, about its Brexit plans.
Speaking on RTÉ's The Week In Politics, Mr Harris said progress had been made last week between the EU and the UK and now "we need to have clarity about what's going to happen next".
The minister predicted that the next 24 to 48 hours would be "very instructive in that regard".
Fianna Fáil's Public Expenditure and Reform spokesman Barry Cowen said he "sincerely hopes" that Mr Johnson will be able to get a deal through.
He told The Week in Politics that the Johnson deal was "not as good as the May deal, but does have certainty for business".
Asked about DUP criticism of the deal, Sinn Féin MEP Matt Carty contended that "a deal that was acceptable to the DUP, wouldn't be agreeable to EU".
Meanwhile, Sammy Wilson of the DUP has tweeted saying that his party's vote supporting the Letwin amendment "is far from the first time that DUP votes have been crucial on the issue of Brexit".
He reiterated that the DUP wants to "get Brexit done" but "it must be a Brexit for the whole of the United Kingdom.
"Our position has been clear and it has been consistent."
He went on to say that "DUP support for the Letwin amendment was a situation that could have been easily avoided had the PM kept to words he penned to Jean-Claude Juncker just a matter of two weeks ago".
He ended by saying that the DUP "does not seek a second referendum; merely implementation of the first.
"The people of the United Kingdom were asked whether the UK should leave the EU, not whether Great Britain should leave Northern Ireland behind.
"Leaving as one nation remains our goal."
Labour MP and Shadow Chancellor John McDonnell has said Mr Johnson is "clearly trying to undermine a decision by parliament" with his latest actions and that he is behaving like a "spoilt child".
Speaking on Sky News, he said by not signing the letter when asking the EU for an extension, he may be in contempt of court.
He also defended the vote taken yesterday in parliament which delays approval of Mr Johnson's deal.
Earlier, Britain's former home secretary Amber Rudd said yesterday's vote was done "to make sure the UK leaves with a deal".
Also speaking on Sky News, she said the vote on an amendment that was passed is an "insurance policy" and just tries to protect the deal that has been struck between the UK and the EU.
She said she support's Boris Johnson's deal and will vote for it and she has told him that.