The Northern Ireland Secretary is holding talks with the political parties to discuss the crisis caused by the Renewable Heat Incentive scandal.
James Brokenshire is meeting Sinn Féin on Wednesday morning, and plans to have talks with the main parties, and the justice minister, within 24 hours.
Martin McGuinness stepped down as deputy first minister in protest at the handling of the botched energy scheme.
Mr Brokenshire said his move means an assembly election is “highly likely”.
The prime minister and taoiseach (Irish prime minister) also pledged to do what they can after speaking by phone on Tuesday night.
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Efforts to try to save the institutions at Stormont and avoid an election are set to intensify in the coming days.
Mr Brokenshire is planning his meetings with the parties.
However, with Sinn Féin insisting an election is inevitable there is little hope the crisis that has brought politics in Northern Ireland to standstill will be resolved anytime soon.
Speaking on BBC’s Good Morning Ulster, Sinn Féin’s Gerry Kelly said the party would not accept any inquiry set up by DUP.
But whether they would accept an inquiry set up by the justice minister or the secretary of state remains to be seen.
James Brokenshire has an eye on the negotiations post-election.
We know a key demand from Sinn Féin was that Arlene Foster step aside while an investigation goes ahead and she is now out of office.
The secretary of state, therefore, may be working on a plan to instigate some kind of inquiry to run along side an election.
On Tuesday night, Mrs May and Mr Kenny agreed the situation is very serious in a 15-minute telephone conversation.
They also asked Mr Brokenshire and Irish Foreign Minister Charlie Flanagan to work together with the parties to find a solution.
Former first minister Arlene Foster warned that if an election goes ahead, it will be “brutal”.
The Renewable Heat Incentive (RHI) scheme could cost NI taxpayers £490m.
Mrs Foster set up the RHI scheme in 2012 when she was enterprise minister, in an attempt by the NI Executive to increase consumption of heat from renewable sources.
However, businesses received more in subsidies than they paid for fuel, and the scheme became heavily oversubscribed.