It comes as a small number of backbench TDs have privately indicated that they would consider withdrawing their support if Mr Kenny does not make the details of his departure known before the end of the year.
Speaking at the party’s think-in in Newbridge, Co Kildare, Mr Kenny said he has “no intention of walking away” from his mandate.
“Stability is a very important element of where we are here. This is not a time to mess around with political parties. Our country faces enormous challenges, unprecedented in fact,” he said.
“This country faces some very serious challenges and I am not, and I have no intention of, allowing destabilisation.
“I have already set out for everybody that I want the entire Fine Gael party and all its members to focus on its principle and on the agenda we have set out.
“We have no room for deviations from that. This is about our people, our country, and our lives. It’s much too important for side matters.”
It is understood that a number of dissatisfied backbenchers would be willing to bring forward a motion of no confidence in Mr Kenny if he does not outline when he will be leaving.
Responding to repeated questioning on whether he would fight such a motion, Mr Kenny said: “I have a mandate from the Oireachtas as Taoiseach and head of government in the most challenging times that our country has seen for a very long time, not just at home but abroad.
Pointing out issues including Brexit, the Apple tax decision, the budget, and the housing crisis, he said: “We have a really busy, challenging agenda up ahead, locally, nationally, and internationally.
“As I have said quite clearly, I am not, and I have no intention of, walking away from my mandate in that regard.”
After next month’s budget, Mr Kenny will likely come under intense pressure to state when he will step aside as party leader.
He was first put under pressure on the matter at a parliamentary party meeting before the Dáil’s summer recess. The issue has rumbled on over the summer months, with a number of Fine Gael members voicing frustrations during this week’s think-in.
Yesterday, Cork South West TD Jim Daly said: “I want to be very clear the reason I am commenting on the leadership of the party is because the Taoiseach himself has said that he will not lead us into the next election.
“That leaves us in a situation where we could have an election at any stage. The instability of this Government is there for all to be seen so I do think it is urgent.”
Mr Kenny also ruled out running for the presidency at any stage in the future, adding that President Michael D Higgins is doing a “first-class job” in representing the country.
“Personally I have no interest other than being an occasional attender as a visitor in Áras an Uachtaráin. I do not wish to become a tenant of the place,” he said.