The Taoiseach Leo Varadkar has reiterated to British Prime Minister Theresa May that the Irish and EU position on the Brexit backstop remain unchanged.
The two leaders spoke by phone on Wednesday afternoon.
EU leaders and negotiators are also bearing down on the backstop - insisting it cannot be opened to any changes.
British MPs voted to reject the backstop on Tuesday night.
The House of Commons backed a move to replace the measure, aimed at preventing a hard border in Ireland, with "alternative arrangements."
Immediately after the vote, the European Council President and the Irish Government again warned that the Brexit Withdrawal Agreement - including the backstop - cannot be re-negotiated.
Despite the impasse, the UK Prime Minister Theresa May is now expected to return to Brussels to seek changes.
Backstop was agreed by UK/EU as the insurance policy to avoid a hard border in all scenarios. We hope it will never be used, or be replaced quickly by a future relationship agreement. But it is necessary and tonight’s developments at Westminster do nothing to change this. #Brexit
— Simon Coveney (@simoncoveney) January 29, 2019
After the vote, Mrs May said her fellow MPs had "now said they would support a deal with changes to the backstop."
“We will now take this mandate forward and seek to obtain legally-binding changes to the Withdrawal Agreement that deal with concerns on the backstop while guaranteeing no return to a hard border between Northern Ireland and Ireland,” she said.
She admitted that "there is limited appetite for such a change in the EU and negotiating it will not be easy," but said the House of Commons had "made it clear what it needs to approve a withdrawal agreement."
In their phone call, Mr Varadkar set out once again the unchanged Irish and EU position on the withdrawal agreement and the backstop.
He noted that the latest developments had "reinforced the need for a backstop which is legally robust and workable in practice."
Mrs May said that further consultations are taking place in London.
Michel Barnier addressing the European Parliament in Brussels | Image: European Union
Earlier, the EU's chief Brexit negotiator Michel Barnier told MEPs in Brussels: "The backstop, as mentioned by President Juncker, is part and parcel of the withdrawal agreement - and this agreement will not be renegotiated".
"The backstop - it's not any kind of dogmatism - it is a realistic solution.
"Throughout the negotiations under your control we sought solutions to the problems created in particular - not only, but in particular - in Ireland by Brexit itself.
"And the backstop as it's included in the withdrawal agreement is the outcome of these extremely intense negotiations over two years involving us with the United Kingdom, never against the United Kingdom."
"Rejecting the backstop as it stands today boils down to rejecting the solution which was found with the British - but the problem remains".
"The European Council and European Parliament on several occasions have already clearly rejected the idea of having a limit in time or leaving unilaterally the backstop, because that would undermine the very idea of the backstop.
"It's an assurance... we don't want to use it will be there if we need that insurance".
It emerged on Tuesday night that the European Commission President Jean-Claude Juncker had warned Mrs May that the EU would not budge on its refusal to renegotiate in a phone call before the vote.
Immediately after, the EU again warned that the backstop must remain in place.
A spokesperson for European Council president Donald Tusk said: "The withdrawal agreement is not open for renegotiation."
"The agreement is a carefully negotiated compromise, which balances the UK position on customs and the single market with avoiding a hard border and protecting the integrity of the EU customs union and single market," he said.
"The best way to ensure an orderly withdrawal is to ratify this agreement."
However, the spokesman did suggest that the EU would consider a "reasoned request" for an extension of the Brexit process.
Here, the Government immediately issued the same statement warning that, "best way to ensure an orderly withdrawal" is to ratify the withdrawal agreement.
It said the EU has consistently said that it wants "the closest possible future relationship" with the UK and noted that a "change in the UK red lines" could lead to a better overall outcome.
"We will continue our preparations for all outcomes, including for a no-deal scenario," it reads.
Speaking at @IFAmedia annual dinner this eve. Brexit is the great political challenge of our time, and we have to hold our nerve. I can’t offer farmers the reassurance provided by certainty, but I can reassure you that until things are certain we will keep fighting your corner pic.twitter.com/3pcUl5c9XD
— Leo Varadkar (@campaignforleo) January 29, 2019
Tánaiste Simon Coveney has also reiterated that the agreement is not open for negotiation.
He told the Institute of International and European Affairs (IIEA) in Dublin: "The withdrawal agreement won't be re-opened, but as the EU has continuously said, the political declaration on the future relationship can evolve if the UK government changes its red lines."
"And if the UK wishes to request an extension of Article 50, the EU27, including Ireland, are ready to consider that request and decide unanimously on it".
French President Emanuel Macron also said the agreement “is the best accord possible” adding that it is “not renegotiable.”
He said a no-deal scenario was one that “no one wants but we should all prepare for.”
The House of Commons voted on a series of amendments on Tuesday night –with Mrs May winning all but one.
MPS voted by 318 to 310 to reject a no-deal Brexit – although the amendment is not legally-binding.
The vote will allow UK Labour leader Jeremy Corbyn to enter talks with Mrs May on a way forward on Brexit.
He had previously refused to meet with the prime minister unless she ruled out a no-deal outcome.
Mr Corbyn told MPs after Tuesday's votes: "Since we have this debate and the House has emphatically voted to reject the no-deal option that the prime minister was supporting, could I say that we are prepared to meet her to put forward the points of the view of the Labour Party of the kind of agreement we want with the EU."
Additional reporting: Jack Quann
