The Taoiseach has said he believes that there is enough "common ground" to allow Northern Ireland's two main political parties to form an executive, however he expressed concern that matters are moving towards direct rule.
Sinn Féin and the DUP failed to reach an agreement in talks on restoring power-sharing in Northern Ireland.
A row between the two parties over an eco-boiler scandal collapsed the power-sharing executive last January and there has been no executive in place since then.
Speaking in Seattle, Leo Varadkar said the Government is not giving up on the possibility that an executive can be formed.
"There's really not a huge gap between the parties now and I certainly encourage them to close that gap and allow the Northern Ireland executive to be established", he said.
"It is really important that Northern Ireland has a devolved government so they can deal with really important issues like health and education for example, putting a budget in place and also ensuring that Northern Ireland has an elected voice when it comes to the Brexit negotiations which currently it doesn't have."
Mr Varadkar expressed concern that Northern Ireland is sliding towards direct rule.
He said the fact that a budget may have to go through Westminster, rather than through Stormont, is in many ways the first step towards a restoration of direct rule.
He said that it is the position of the Irish Government which cannot support a return to direct rule in the form that existed prior to the Good Friday agreement.
Earlier, Minister for Foreign Affairs Simon Coveney said the failure of Sinn Féin and the DUP to reach an agreement is a step towards direct rule.
Speaking on RTÉ's News at One Mr Coveney said it is "frustrating" that Sinn Féin and the DUP were unable to strike a deal, but he said he believes there is "the basis for an agreement" and that significant progress was made in this phase of negotiations.
Minister Coveney said he "does not accept" that today's outcome brings forth de facto direct rule, but that "it is a step towards decisions for Northern Ireland not being made in Northern Ireland."
Mr Coveney said he believes that there is the "basis of an agreement" between the two largest parties in Northern Ireland, but that both Sinn Féin and the DUP need to be willing to compromise.
He also said he believes Ireland has a role to play in Northern Ireland if power-sharing is not restored.
Mr Coveney said both Sinn Féin and the DUP must be willing to compromise.
"It is certainly a step in the direction of direct rule, which is certainly not a place we want to be in," he said.
He added: "We've made it clear that if we move towards direct rule, if we can't find a way of getting devolved government in place, then we will have to fall back on the full detail and implementation of the Good Friday agreement in terms of the structures that are required in the absence of the devolved government."
He said the Irish government will have to fulfil its obligation as co-guarantor of the Good Friday agreement, and the clear structures within it on how Northern Ireland should be governed in the absence of devolved government.
He said it will "be no easy glide path to direct rule, it will be a very bumpy ride", but that the two governments need to work out together how each can perform their legal obligations under the agreement.
Mr Coveney added that neither government wants to see a situation where Northern Ireland is not making decisions for itself.
"There is in my view the basis of an agreement in Northern Ireland to be struck but it requires the political will by the two largest parties to do that. For the other parties in Northern Ireland, they have shown extraordinary patience."
He said the Secretary of State James Brokenshire is right to address the issue of paying MLAs.
Minister Coveney said it is "not sustainable to continue to pay people full salaries when they're not doing a full job."
DUP will continue to engage with Sinn Féin - Foster
DUP leader Arlene Foster has said she will continue to engage with Sinn Féin to try and find a solution that would restore power sharing in Northern Ireland.
Foster says DUP will continue to engage with Sinn Féin on trying to find a solution pic.twitter.com/okk0EGxmzv— RTÉ News (@rtenews) November 1, 2017
Speaking to RTÉ News, Ms Foster said she wants an agreement that unionists and nationalists are comfortable with, and she is "committed to finding a way forward and bringing back devolution to Northern Ireland."
She said Sinn Féin's preconditions imposed since they left government in January but that she remains optimistic that a deal can be reached.
Ms Foster said some of the significant members of her team were absent this week due to commitments with their families over half-term.
She said this has not affected the party's engagement with Sinn Féin.
On the issue of the Irish language, Ms Foster said: "The St. Andrew's agreement referred to 'an' Irish language act, not the Irish language act" and she said this is a "significant difference."
She said her party did not sign up to this particular point that was put in to the agreement by the then British government.
The DUP leader said her party is trying to find common ground where agreement can be found for unionism and nationalism.
Earlier, Sinn Féin's leader in Northern Ireland, Michelle O'Neill, said that talks at Stormont "have ended in failure" due to the inability to "make sufficient progress in terms of delivery of rights".
However, she said the party was willing to re-engage at any time, subject to the delivery of rights.
She added that the dialogue had to continue if an executive was to be formed at Stormont.
Party leader Gerry Adams blamed the British government and the DUP for the failure of the talks, saying Sinn Féin "want to see a rights-based society".
'The issue of rights is not going to go away' – Sinn Féin say it is clear negotiations in Northern Ireland have ended in failure pic.twitter.com/s4BW1KDXD4— RTÉ News (@rtenews) November 1, 2017
A senior member of the DUP, Gregory Campbell, blamed Sinn Féin for the collapse of talks, saying they were holding the government back with a "narrow political agenda."
He also suggested the DUP would not compromise on the key issue of the Irish language.
Earlier this week, Sinn Féin and the DUP failed to meet a Monday deadline to reach agreement to reinstate the Stormont Executive.
Northern Secretary James Brokenshire had previously warned the parties that they had until the start of the week to produce a written agreement or he would be forced to legislate for a budget for the region at Westminster.
As a result the British government has said it will begin the process of directly imposing an annual budget for Northern Ireland after efforts to restore power-sharing at Stormont proved unsuccessful.
However, Mr Brokenshire said talks between the parties would continue and the budget process could be handed back if an executive was formed.
James Brokenshire to introduce legislation to allow for budget in Northern Ireland pic.twitter.com/mriaWk9rmP— RTÉ News (@rtenews) November 1, 2017
"While important progress has been made in discussions between the DUP and Sinn Féin towards the establishment of an executive, it has not yet been possible for the parties to reach agreement," Mr Brokenshire said in a statement.
"I am, therefore, now taking forward the necessary steps that would enable a budget bill to be introduced at Westminster at the appropriate moment in order to protect the delivery of public services in Northern Ireland."
A spokesman for British Prime Minister Theresa May's told reporters that Mr Brokenshire would continue to facilitate talks between the parties, saying: "Our aim is still to get to a position where the Assembly can get up and running again. That is our priority."
Asked whether the imposition of a budget would mean Northern Ireland losing out on the £1 billion of government investment agreed at the time of the DUP's deal to prop up the minority Conservative administration, a senior Downing Street source said: "The money that has been promised as part of the confidence and supply deal is for the people of Northern Ireland and we are committed to delivering on that undertaking."