The kingmakers: five key ministers who could influence leadership race
On Friday, Education Minister Richard Bruton pointedly refused to rule out a run for the leadership when asked by the Irish Independent.
His intentions will become clearer tonight when he attends the AGM of his constituency branch in Dublin Bay North.
It is understood a number of delegates intend to seek confirmation from Mr Bruton over whether he will throw his hat in the ring.
Sources say that while he is unlikely to secure a great deal of support from the parliamentary party, he believes he will benefit from a strong popularity among the Fine Gael grassroots.
But Mr Bruton will be conscious of the need to steer clear of any motions of no confidence in Mr Kenny - having led an unsuccessful heave against the Mayo politician in 2010.
In the eyes of senior Fine Gael figures such as Michael Noonan and Michael Ring, Paschal Donohoe is the ideal candidate to succeed Enda Kenny as Taoiseach.
He has shown himself to be a solid pair of hands at the Cabinet table and would be well-placed to unify the party during the post-Kenny period.
But Mr Donohoe has made it clear that on this occasion, he will not be putting his name forward.
He is not the type of politician to go back on his word and therefore his bona fides should be trusted.
But Mr Donohoe has said he will eventually endorse a candidate - rather than sit on the fence. His endorsement could prove significant.
As soon as the leadership issue was raised last week, Tánaiste and Justice Minister Frances Fitzgerald told her inner circle that she was interested.
But the prospect of the country electing its first female Taoiseach has gained little traction within the Fine Gael ranks since.
This is due to her botched handling of the whistleblower affair and her embarrassing 'he says, she says' row with Fianna Fáil's justice spokesperson Jim O'Callaghan over the Tusla affair.
Ms Fitzgerald does have supporters in the party, however, and any endorsement she offers could prove significant. She may yet decide to back Health Minister Simon Harris, who is seen as her protégé.
Finance Minister Michael Noonan is widely expected to be the first casualty from Cabinet when the new leader is elected in the coming months.
However, the Limerick politician may end up playing a crucial role in advising one of the leadership hopefuls.
He has already admitted that he would like to see Paschal Donohoe become leader. But whether he endorses a candidate publicly remains to be seen at this stage.
While it is accepted that Mr Noonan's career is in its latter stages due to his poor health, grassroots members are likely to be highly influenced by the veteran politician.
Although aged just 30, Health Minister Simon Harris is himself being speculated as a potential 'compromise' candidate.
He spent the weekend consulting supporters in Greystones, Co Wicklow, and is said to be seriously considering entering the race once Enda Kenny makes his intentions known.
Central to Mr Harris's thinking is positioning himself for a further tilt at the leadership in the future.
If he does perform well in any contest, he will guarantee himself a senior ministry in either Leo Varadkar or Simon Coveney's cabinet.
He is likely to announce his approach following negotiations with Tánaiste Frances Fitzgerald, whom he is closely aligned to in the parliamentary party.