Latest: DUP may call for direct rule in House of Commons debate
It is thought the DUP may make a call in the British House of Commons later, for the return of direct rule in the North.
Party leader Arlene Foster says her MPs will be raising the issue later.
It comes as the Northern Ireland Secretary Karen Bradley is due to update the British House of Commons on how the UK Government will deal with the situation.
The Irish and British governments have insisted they are fully committed to the Good Friday Agreement.
Taoiseach Leo Varadkar and British Prime Minister Theresa May spoke by phone after the Democratic Unionists and Sinn Féin clashed over the prospect of direct rule being imposed on Northern Ireland.
Both leaders expressed disappointment over the political impasse at Stormont.
The breakdown in powersharing came to a head despite optimism that a deal had been close on contentious issues such as the Irish language, marriage equality and the legacy of the past.
"Both leaders recognised the progress and serious engagement made by the parties," a spokeswoman for 10 Downing Street said.
"The Prime Minister said she believed there was scope for agreement and reiterated the UK Government's priority was still to get devolution up and running again in Northern Ireland."
Mrs May and Mr Varadkar are to remain in close contact.
The leaders spoke after the Taoiseach and his deputy, Foreign Affairs Minister Simon Coveney, met Sinn Féin leader and deputy leader Mary Lou McDonald and Michelle O'Neill at Government Buildings in Dublin.
A spokesman for the Taoiseach said he emphasised the Irish Government's full commitment to the Good Friday Agreement and its determination to secure the effective operation of all its institutions.
"As co-guarantor to the agreement, the Government will continue to engage with the parties in Northern Ireland and the British Government to support the urgent formation of a new executive by the mandated political parties," the spokesman said.
The Taoiseach's office said the Good Friday Agreement and subsequent agreements must be implemented in full and reiterated opposition to direct rule.
Earlier Democratic Unionist leader Arlene Foster had called on the UK Government to start setting budgets for schools and hospitals.
"Frontline staff have been living hand to mouth for too long. The Northern Ireland people deserve better," Ms Foster said.
"I am not prepared to allow this to continue. Decisions need to be taken."
Mrs McDonald dismissed the DUP calls for Westminster to start making big spending decisions in Northern Ireland.
"We don't accept any call for a cooling-off period, in fact we're saying very clearly that there cannot be vacuum, we need action and we need the two governments to be decisive at this point," she said.
"Direct rule is not acceptable. Direct rule is not on the table.
"We have been clear - nationalism right across the country is clear on that point, the Government in Dublin is clear on that point and I have to say that we have previous words from both governments which made very clear that direct rule is not the answer when the institutions falter."
Sinn Féin is pressing for the British Irish Intergovernmental Conference to be convened in the absence of powersharing.
The Sinn Féin leadership and Ms Foster are expected to hold separate talks later this week with Prime Minister Theresa May.
The DUP, whose 10 MPs are propping up the Conservatives at Westminster, effectively ended talks aimed at bringing the 13-month political impasse in Northern Ireland to a close last Wednesday when Ms Foster said she did not believe it would be possible to form an executive.
The move came two days after the British PM visited Stormont.
