Tánaiste Frances Fitzgerald, Health Minister Simon Harris, and Education Minister Richard Bruton are all still understood to be in the mix to run for leader of Fine Gael.
It is understood that Mr Harris is still considering his own options and may put his name forward as a third candidate in the race.
However, the Wicklow TD, who has been a loyal supporter of Ms Fitzgerald, would be unlikely to put his name forward if it meant campaigning against the Tánaiste.
Ms Fitzgerald has long hinted at her ambition to lead the party.
However, she has been significantly damaged this week through her handling of the McCabe case.
It is understood that Ms Fitzgerald is still considering her options but is also very conscious of the fact that there is, as yet, no vacancy and Enda Kenny has to be given the chance to announce the timing of when he will leave.
Party sources have also claimed that Fine Gael stalwart Richard Bruton cannot be ruled out of putting his name forward, but he has so far remained silent.
While Simon Coveney and Leo Varadkar are seen as the two frontrunners to succeed as leader of Fine Gael, those loyal to Mr Kenny may decide to put forward a candidate.
A senior Fine Gael source said: “There is the Enda Kenny wing of the party and they might go to Frances [Fitzgerald], they might go to Simon [Harris] or to Paschal [Donohoe] if they are to run. This will be a big change whenever it happens, it has been 15 years since there was last a change of leader.
“So a lot of people are still unsure and are weighing up their options.”
The source added: “What was clear this week is an election can happen at any time, Enda Kenny has said he won’t be leading the party into another election, so the issue of leadership has to be addressed.”
Meanwhile, Minister of state for the Diaspora, Joe McHugh, echoed calls to give the Taoiseach space to make his mind up on his future.
He told RTÉ RnaG: “The right thing to do is to give Enda Kenny space and time to make his decision. It’s Enda’s decision.”
Asked if there is a danger of an early election he said: “Things have changed in the last week and an election is nearer than it was before last week.
“Enda is very intelligent and he has been all over the country since 2002 when he came into the job first. He has done an excellent job for the party,” said Mr McHugh.
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