Before the summer recess he came under intense pressure from a number of back-bench TDs to outline the timing of his departure and this discontent has rumbled on over the summer.
While many Fine Gael TDs and senators publicly say they will support Mr Kenny for as long as he wants to stay, there is a significant group who believe he should stand down sooner rather than later.
Speaking at the Fine Gael think-in in Newbridge yesterday, Carlow-Kilkenny TD Pat Deering said the leadership issue will have to be resolved by Christmas time.
But Kate O’Connell voiced support for the Taoiseach by stating “the leadership issue is a matter for Enda Kenny, he is the leader”.
She added: “It’s his call. It’s not up to me to tell Enda Kenny when he needs to do anything, it’s up to him, so it will be his choice.
“I don’t think we are looking at a general election any time soon anyway, I haven’t the posters cleaned.”
It comes after Kerry TD Brendan Griffin told the Irish Examiner that Fine Gael was on “high alert” for a snap election but added there was the “huge issue of our leadership, who leads us into the next election”.
Yesterday one backbench TD said many in the party wanted the leadership issue resolved.
“It will have to be dealt with over the next recess.
“A new leader would need around a year to settle in before any election so they can settle in and are ready to lead the party into another election.”
The TD said he did not see any immediate threat to Mr Kenny’s position, but added that members would become restless if he did not stand down before the new year.
Another TD, who said he supported Mr Kenny as leader, did admit that there was division in the party, especially on the backbenches, with many still unhappy with the current leadership.
Dublin North-West TD Noel Rock said: “There will always be a few disaffected people, a few unhappy apples in the barrel.”
However, he added that there was “no impetuous” for Mr Kenny to put a timeline on leaving.
Dublin South-West TD Colm Brophy said: “He is the party leader and he can remain the party leader for as long as he wants.
“People are not doing timelines, people are working on the basis that he has said that he has a job of work to do as Taoiseach which we fully support him, he can take as long as he likes to do that and then he will step down and he will indicate that to the party at a time that he wants to.”