Several TDs also took to the airwaves yesterday and openly criticised the party direction after a disastrous week for Mr Kenny was topped off with dismal poll results for his party.
Senior Fine Gael figures are now admitting that a discussion on who takes over the leadership is inevitable and it is just a matter of when and how.
Fine Gael TDs Jim Daly, Pat Deering, Michael D’Arcy and Peter Fitzpatrick openly spoke of leadership change yesterday, after a heated party meeting in Leinster House on Wednesday night.
Mr Daly said he wants Mr Kenny to reveal his intentions on Fine Gael’s leadership after October’s budget. Mr Kenny has said he will not lead his party into the next general election.
Mr D’Arcy said his party was currently “drifting” and the ball was now “rolling faster than anyone” had anticipated.
Mr Deering questioned why a leadership change needed to be a “drawn-out process”.
TDs also expressed surprise that senator James Reilly was reappointed as deputy Fine Gael leader at the Wednesday night meeting.
Party chairman Martin Hayden said Mr Kenny was the best person at present to lead the country and negotiate the best deal for Ireland post-Brexit.
It was difficult for his party leader to set a date on when a new leader would be sought as this may destabilise the party, he said.
But Housing Minister Simon Coveney was candid when he admitted “there will be a time in the not too distant future I expect when leadership is going to have to be discussed in Fine Gael”.
Mr Coveney said there were issues in the party that needed to be resolved internally and that “there was an acceptance that this week hasn’t been a great week for the Government”.
The minister said the issue of the party leadership was not his “focus right now”. However, he also admitted that he had not known about the reappointment of Mr Reilly as deputy leader.
Elsewhere, Social Protection Minister Leo Varadkar confirmed he would “love to lead” Fine Gael.
But Mr Varadkar, seen as a strong contender to take over from Mr Kenny, also said there was no vacancy for the job at the moment.
“Of course I’d love to lead my party — I’ve been dedicated to it since I was 17 or 18 years old,” he said.
“But that’s something you have to consider at the appropriate time, when a vacancy arises”.
The trouble for Mr Kenny comes as an MRBI poll yesterday put Fine Gael at 24%, down 2%, but gave Fianna Fáil an increase of 9% to 33%.
Fine Gael party officials last night suggested that Mr Kenny seemed very “isolated” at present, especially given his main advisors were on holidays and his chief economic advisor was set to leave.
“These are self-inflicted injuries by his own hands in the last week. This [criticism] is not coordinated, it’s an organic outpouring of frustration,” said one inside source.