The Taoiseach and British Prime Minister Theresa May have spoken by telephone this afternoon, where Leo Varadkar reiterated the firm Irish position regarding the text of the Brexit deal as outlined by him on Monday.
Mr Varadkar and Mrs May took stock of developments on the issue since Monday, and they agreed to speak again over the coming days.
One idea mentioned in the conversation was the possibility of adding a clause to say any agreement was not about the break-up of the United Kingdom.
No wording has yet been forwarded to Dublin via the European Commission Task Force, but it is expected arrive here soon.
However, the Taoiseach has warned that any change must not alter the substance of what was almost agreed last Monday.
And that means getting a guarantee that a hard border will be avoided once the UK leaves the European Union.
Mr Varadkar said the rules do not have to be identical on both sides of the border, but sufficiently aligned to allow for continued north-south cooperation.
The Government got full support for its position from the Prime Minister of the Netherlands Mark Rutte.
Earlier, the Taoiseach said that if it is not possible to begin Phase 2 of the talks to address post-Brexit trade at next week's EU summit "then we can pick it up in the New Year".
He acknowledged it was in Ireland's own interest to see the EU-UK negotiations proceed to their second phase once the European Council meets on 14-15 December.
He added: "I think we should listen to all parties in Northern Ireland and not accept this idea that seems to be gaining prevalence in some parts of London and maybe other places as well that there is only one party in Northern Ireland and that party speaks for everyone.
"I don't accept that premise, which seems to be accepted by too many people at the moment."
The Taoiseach said it was the UK's role to come back to the EU side with proposals.
"I understand that the Prime Minister has difficult issues that she is managing, there are different views within her own party on Brexit and she also has to manage a confidence and supply agreement that she has with the DUP.
"I absolutely accept that Theresa May wants to come to an agreement, that she is acting in good faith and I want to give her time ... before we move things forward."
He said it was the "desire, ambition and wish" of the Government to move on to post-Brexit trade talks.
"It is in our interest to move to Phase 2, that is where we talk about the transition period that we need, so individuals and businesses can prepare for any long-term change.
"It is where we can talk about the new trading arrangements which are so important for Irish importers and exporters, the agri-food industry and anyone whose job in Ireland depends on trade with Britain."
In a press conference this evening, Leo Varadkar said that there must be assurance that any new UK language is consistent with the agreed border text but none has been proposed yet.
He said that he would examine any text put forward in positive light and with an open mind.
However, he said he has little room to manoeuvre on what was outlined last Monday.
Meanwhile, the DUP has warned that a stand-off over the wording of an agreement on the future of the border between Northern Ireland and the Republic increased the likelihood of a "no deal" Brexit.
The party’s deputy leader, Nigel Dodds, said that Mr Varadkar was playing a "dangerous game" by suggesting the border agreement might not be secured until next year, delaying Brexit trade talks.
In a statement, Mr Dodds said: "The longer there is delay in getting onto the second phase of the negotiations about a trade deal, the greater the prospect of a 'no trade deal' outcome."
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Earlier, Mrs May pledged to protect the "constitutional integrity" of the UK in the negotiations to break the deadlock in the Brexit talks over the border between Ireland and the UK.
Mrs May, who spoke earlier by phone to Democratic Unionist leader Arlene Foster, said her government was committed to ensuring there was no return to a "hard border".
However, she told MPs at Prime Minister's Questions that the issue could only be finally resolved once negotiations with the EU move onto the second phase - including talks on a free trade deal.
Her comments came after the DUP scuppered a deal on Monday to enable the talks to move the second phase, saying they could not accept the government's proposal that there should be continued "regulatory alignment" between the North and the Republic.
Ms Foster said it amounted to imposing a "border in the Irish Sea" between Northern Ireland and the rest of the UK.
Mrs May said: "We will ensure that there is no hard border between Northern Ireland and the Republic of Ireland.
"We will do that while we respect the constitutional integrity of the United Kingdom and while we respect the internal market and protect the internal market of the United Kingdom.
"That is the point of the second phase of the negotiations, because we aim to deliver this as part of our overall trade deal with the European Union and we can only talk about that when we get into Phase two."
A spokesman for Mrs May warned against the idea that discussions with the DUP to resolving a Brexit row could involve the Northern Irish budget.
"I'd warn you against speculating in that direction," the spokesman told reporters when asked if talks involved financial issues and Northern Ireland's budget.
He said the government would not be providing further details of a telephone call between Mrs May and Ms Foster earlier in the day.
Meanwhile, Sinn Féin's leader in Northern Ireland has said she made clear to the Mrs May that the DUP does not represent the views of the majority of people in the North of Ireland.
Speaking on RTÉ's News at One, Michelle O'Neill said the DUP seem to have "set their face for a hard border and this is unacceptable."
She said she believes special status is necessary for Northern Ireland post Brexit and any Brexit text would have to reflect its special status.
She said the wording which had been proposed earlier this week in relation to a potential agreement on the border issue "cannot be diminished any further", and that it is the "absolute minimum" that can be agreed for the people of this island.
Ms O'Neill said the ball is in Theresa May's court, adding that she has to stand by what has been agreed to date with the Irish Government.
Earlier, a spokesperson for Mr Varadkar said: "The view of the Irish Government is that the terms of the deal reached on Monday must stand. It's up to the UK government to work out with the DUP how it now proposes to move forward," a spokesperson for the Taoiseach has said this morning.
He was commenting on reports that a clarification line could be added to the text of the agreement that would allow the Brexit talks to move to Phase 2.
Minister for Finance Paschal Donohoe said the text agreed between the UK and the EU on the border between Ireland and the UK "must be retained".