Taoiseach Enda Kenny discussed the Stormont executive with British Prime Minister Theresa May in a phone call this afternoon.
After a snap election radically altered the face of the Stormont Assembly, abolishing for the first time the overall unionist majority, political leaders have three weeks to form an executive.
But the two main parties, the Democratic Unionist Party and Sinn Féin, are on a collision course over Arlene Foster's leadership.
Mr Kenny and Mrs May have held a 15-minute telephone conversation about the election outcome.
The pair agreed that early engagement by the political parties in Northern Ireland is now required with a view to re-establishing a functioning executive as soon as possible, said a Government spokesperson.
Mr Kenny and Mrs May also agreed that Minister for Foreign Affairs Charlie Flanagan and Secretary of State for Northern Ireland James Brokenshire would engage together with the parties over the coming days, added the spokesperson.
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The two leaders agreed to discuss the issue again at the EU council summit in Brussels on Thursday.
Earlier, Mr Flanagan said the Irish and British governments would work closely with the political parties in Northern Ireland to try to ensure the Stormont Assembly and Executive was restored.
Sinn Féin have refused to pull back from its red line that the DUP leader can not be reinstated as first minister while an inquiry is ongoing into alleged corruption and misuse of public money in a heating scheme scandal that forced last week's snap poll.
The DUP has insisted Sinn Féin cannot dictate who they nominate to lead the party in any restored Stormont Executive.
Mr Flanagan, who has spoken with Mr Brokenshire, said the two governments were agreed on the need for intensive engagement to address outstanding issues and ensure the early resumption of the Executive.
Mr Flanagan said it was of the utmost importance for the people of Northern Ireland that the political institutions, established under the Good Friday Agreement, promptly resume "not least so that they can effectively engage with the issues raised by Brexit".
Mr Brokenshire said responsibility lies on the shoulders of both the DUP and Sinn Féin.
The Secretary of State added "confidential" talks would start immediately to resolve other outstanding issues over the full implementation of peace agreements in the region and how the legacy of the Troubles is addressed.
DUP 'not ruling out' Foster stepping aside temporarily
Democratic Unionist Party MP for East Belfast Gavin Robinson has said the party is not ruling out its leader Mrs Foster stepping aside temporarily.
Speaking on the BBC's Five Live, Mr Robinson said it would be Mrs Foster's decision to take and one she would make herself, and the party would consider it.
Mr Robinson acknowledged that the election campaign was a difficult one for unionism and said it was not the result unionists would have liked. He said the election was about "the selfish interests of Sinn Féin" who campaigned on equality, which he said was a "Trojan horse" to break unionists.
He said the greater desire of the people is to see a return to devolution in Stormont on a basis where it can work for Northern Ireland.
He denied that the DUP had any "red line issues" in government and he said Ms Foster's stance that she would never accept the Irish Language Act was a long-held policy position.
Mr Robinson said he has not detected any moves against Ms Foster from within the DUP and despite it being an "extraordinarily difficult period" the party has stuck with her and the party had "a good election" across the province.