Theresa May phoned Northern Ireland's First Minister Arlene Foster and Sinn Fein's Martin McGuinness early on Monday in a last-ditch effort to prevent the collapse of the devolved administration.
The Prime Minister's official spokeswoman said Mrs May wanted to make sure Northern Ireland has "a voice" in the run-up to the start of EU withdrawal talks - expected to be triggered by the end of March with the invocation of Article 50 of the European Union treaties.
If Sinn Fein refuses to nominate a deputy first minister to replace Mr McGuinness by 5pm on Monday, the institutions will fall and Northern Ireland Secretary James Brokenshire will be legally obliged to call a snap election.
Mr McGuinness dramatically resigned his post last week amid the row over a botched green energy scheme.
Mrs May's spokeswoman told a regular Westminster media briefing: "She spoke to both Arlene Foster and Martin McGuinness to encourage them to use what time was left today to try to find a resolution to the situation in Northern Ireland, outlining the importance of finding a way forward - particularly in the context of wanting to make sure that Northern Ireland has a voice in the UK's exit from the European Union as we approach the critical period before triggering Article 50."
The spokeswoman said the Government hoped to continue with planned meetings of the Joint Ministerial Committee - which brings together ministers and officials from the UK Government and the devolved administrations - to discuss Brexit, even if Stormont is plunged into elections.
The next scheduled meeting of a JMC sub-committee is due to take place on Thursday.
The PM's spokeswoman said political stability would give Northern Ireland "a greater opportunity to have its voice heard" in the Brexit debate.
"We want to do all we can to make sure that the political stability in Northern Ireland, which was hard-earned, is not thrown away," she added.
A senior Stormont civil servant has warned a long period without government in Northern Ireland will leave the region facing serious financial difficulties.
An election is not expected to solve a bitter dispute at the heart of powersharing between Sinn Fein and the Democratic Unionists and if those parties again emerge as the two largest in the wake of the poll, an immediate return to power-sharing is unlikely.
A major talks process would likely be required, raising the spectre of devolution being suspended and direct rule from Westminster being reintroduced.
The Stormont meltdown has torpedoed the process of agreeing a budget for the next financial year. Without a budget or ministers, civil servants would take control of the purse strings, but would be limited on the money they can spend.
Giving evidence to the Assembly's Finance Committee, Department of Finance permanent secretary David Sterling said: "A long period without a government would be difficult for us to manage.
"Our objective as civil servants would be to ensure minimal disruption...but I wouldn't want to downplay the difficulties. It will be difficult. I am hopeful it will be as short as possible an interruption to government."
Mr McGuinness's move was precipitated by the renewable heat incentive (RHI) scandal - a botched eco-scheme set to cost Stormont £490 million - but the row has also reignited a range of other disputes dividing the DUP/Sinn Fein-led coalition.
The reappointment process is one of a number of elements of the political crisis due to be raised in the Assembly chamber on Monday.
Emergency proposals aimed at reducing the RHI overspend will be tabled by DUP Economy Minister Simon Hamilton while Sinn Fein will also table a motion of no confidence in DUP Speaker Robin Newton.
Mr Newton has been under political pressure over his handling of a recalled Assembly session to debate RHI before Christmas.
He has also been forced to defend himself against conflict of interest accusations in regard to his handling of Assembly exchanges on a controversial charity in his east Belfast constituency.
The devolution meltdown has cast a shadow of uncertainty over a series of big ticket Stormont Executive plans.
One of those is a payment scheme for households losing out under the UK Government's so-called "bedroom tax" and, also on Monday, DUP Communities minister Paul Givan will bypass the Executive to ask for direct Assembly approval for the support measure.
Mr McGuinness's resignation automatically removed DUP leader Arlene Foster from her position as first minister - as executive structures dictate one cannot govern without the other.
On Sunday, Mr Brokenshire said no alternatives to power-sharing were being contemplated, which could include direct rule or joint authority between London and Dublin.
Sinn Fein has accused the DUP of "arrogance and disrespect" in office and insisted republicans will only enter another executive if its long-term coalition partners give way on a series of "equality issues", such as the Irish language and gay rights.
The DUP, for its part, has made clear it will not deliver a "republican agenda" and said the very structures of mandatory coalition powersharing need to be reviewed before another executive is formed.
Ahead of the key Assembly session, Mrs Foster said the electorate did not want or need an election.
She accused Sinn Fein of forcing an election.
"They have forced an election that risks Northern Ireland's future and stability and which suits nobody but themselves," she said.