The Taoiseach has refused to discuss any plans for his departure, saying he is focused on his job as leader of the country.
Enda Kenny's defiant stance may set up a confrontation with his own TDs, who are demanding he give a timetable to step down.
Fine Gael's vice-chairman Pat Deering has threatened a motion of no confidence next week if Mr Kenny has not announced a departure plan.
But this evening, Mr Kenny refused to entertain any talk of retirement, and said he had plenty on his plate.
"I'm focusing entirely on a really busy and challenging agenda up ahead," he said.
"We've got lots of work for all of our party to engage themselves in, ministerial portfolios and programmes for Government to implement…That's what we should be about."
Leo Varadkar has been installed as the bookies favourite to succeed him, the Social Protection Minister says Mr Kenny will know when the time is right to depart.
But backbench TDs say a timeline for Mr Kenny to leave will have to be spelled out in the coming days.
Leo Varadkar, described Mr Kenny's leadership as "fabulous".
But he added: "The events of this week showed a timeline for a general election is now shorter than we might have thought.
"The Taoiseach has said he won't lead us into the next general election and I've absolutely every confidence that he will know when the right time is for himself and the party and the country to step aside."
Another of the frontrunners to replace him is Simon Coveney, Housing Minister, while rising star Simon Harris, the Health Minister, has also been named in some circles as a potential candidate.
Finance Minister Michael Noonan said another election was closer but added that Mr Kenny has a "skill set that is very valuable for the country".
Henry McKean of NewsTalk travelled to Enda Kenny's hometown of Castlebar to hear what people in his hometown think about the pressure on him to step down as Taoiseach.