Taoiseach Leo Varadkar has said the Government's "focus is on securing an agreement" on Brexit.
Speaking at a news conference with Danish Prime Minister Mette Frederiksen in Copenhagen, he said he believed it was possible to do so, but that all sides had to "reaffirm the shared objectives".
He said that the objectives were "no hard border between Ireland, north and south, that the integrity of the Single Market of the European Union will be protected, and that the all-island economy will be protected.
Mr Varadkar added that all sides need "to refocus on those objectives and try to come to an agreement by the middle of October, which I think is possible".
The Taoiseach said that as a consequence of Brexit, Ireland has "always acknowledged there would have to be checks regrettably and we've argued that the least intrusive way to do that is to have those checks in the ports and airports, rather than along a 500km border with 300 crossings.
"The agreement we came to with Prime Minister (Theresa) May satisfied that ... the difficulty with what Prime Minister (Boris) Johnson has put on the table is that on the face of it at least it appears to create two borders".
Mr Varadkar also said that Ireland would be open to a request from the UK for an extension to the Brexit process.
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Meanwhile, Tánaiste and Minister for Foreign Affairs Simon Coveney has insisted he believes a Brexit deal can be reached before the approaching 31 October deadline.
He denied it was "mission impossible" as he left a meeting with Northern Ireland Secretary Julian Smith at Hillsborough Castle.
Reiterating earlier comments from the Taoiseach, Mr Coveney said if an agreement was not possible this month, then Ireland would prefer an extension to a no-deal exit.
"I believe it is possible to get a deal this month," he said.
"I believe that the British Prime Minister wants to get a deal this month. We will certainly work to that time frame if the British position evolves and we want to be helpful on that.
"I don't think we should give up on that.
"If it's not possible Ireland has always said we would prefer an extension to no deal.
"But that's the Irish position but I don't speak for lots of other EU member states."
'This is not mission impossible. There is a deal to be done here' - Tánaiste @simoncoveney #Brexit | Latest headlines https://t.co/3EeanbNFVk pic.twitter.com/JaKpbmnmSM
— RTÉ News (@rtenews) October 4, 2019
Mr Coveney left Hillsborough Castle as DUP leader Arlene Foster arrived for her party's meeting with Mr Smith.
Yesterday, Ms Foster accused the Government of trying to "ride roughshod" over unionism, as she criticised Dublin's rejection of the UK's latest Brexit plan.
Mr Coveney said he would not be getting into a personalised row, and that Ireland's position had not changed.
"I'm not getting into personalised commentary like that, the Irish Government has been consistent for three years now.
"While we respect the decision of the UK to leave the EU, we also expect the British government would take account of Irish interest, vulnerabilities and exposure and the context of those issue of any Brexit deal that is struck," he said.
Meanwhile, Minister of State for European Affairs Helen McEntee has said the proposed Brexit plan could provide the basis for further discussions, but does not currently address some key concerns relating to Northern Ireland.
Speaking on RTÉ's Morning Ireland, she said the Government has been "very clear" about these concerns, particularly around the proposed principle of consent as it's set out in this document.
