Main parties have met Leo Varadkar for talks in Dublin
The leader of the Democratic Unionist Party (DUP) Arlene Foster says she held a "very useful and pleasant meeting" with Taoiseach Leo Varadkar.
The DUP and Sinn Féin have held separate talks at Government Buildings in Dublin over forming a Northern Ireland power-sharing executive at Stormont.
The two parties' discussions comes as the DUP looks set to enter a deal with the Conservatives in Westminster.
British Prime Minister Theresa May will depend on the unionist party to support her government, after her party lost their majority in last week's election.
Concerns have been raised - including by Mr Varadkar himself - that the setup could jeopardise the Northern Ireland peace process.
Sinn Féin's delegation was led by Gerry Adams, while the party's leader in the North Michelle O'Neill also attended.
The Government say there was agreement on the need to re-establish the Northern Ireland Executive urgently - noting the deadline of June 29th.
There was also agreement on the importance of a functioning executive in the context of the Brexit negotiations, which are due to start next week.
Taoiseach Varadkar made clear that he and the Government would work in support of the Northern Ireland parties to re-establish the executive, and for strong North-South relations.
He stressed the objectives of ensuring that Brexit does not impact negatively on the Good Friday Agreement and the peace process, and protecting North-South trade and economic activity and the Common Travel Area.
Speaking after the talks, Mrs Foster said she wanted to see devolution back up and running in Northern Ireland "as quickly as possible".
"I think it is very realistic that a deal can be done, I think the issues that have to be dealt with have been talked about now for quite a period of time... I think it is very much doable to have a deal by the end of this month", Ms Foster told reporters.
Sinn Féin's leader in Northern Ireland Michelle O'Neill says they remain "fully committed" to making the institutions work.
"We need this new Taoiseach to have a new approach, to have a step change in their actions in relation to securing implementation of previous agreements.
"We want to make the institutions work, the people have given us a mandate to make them work.
"This Taoiseach needs to step up, he needs to make sure that he plays his role and takes it very seriously - that role of co-guarantor".
Speaking earlier on Newstalk Breakfast, Sinn Féin's deputy leader Mary Lou McDonald said that they hope to resume talks in the North on Monday.
She said: "We have work to do. We want to get the institutions re-established.
"We were in with Theresa May yesterday. We've set out very clearly the fact that her government is in default of the Good Friday Agreement and successor agreements.
"We've also raised concerns around any side deal or understanding with the DUP that could undermine the agreements and make the job of re-establishing the institutions even more difficult."
She added: "It is good for the North, and it is good for all of us, to have the power-sharing institutions working. That means we have to finish out and implement things that have already been agreed."
Additional reporting: Jack Quann