Kerry Fine Gael Cllr Bobby O’Connell said there are fears among the party’s grassroots that an early General Election could leave the party exposed.
A number of backbench TDs appeared to challenge Enda Kenny in the Dáil this week.
Minister Simon Coveney said he’d like to see the party’s leadership looked at in the “not too distant future”.
Meanwhile, Minister Leo Varadkar said he’d love to become leader of Fine Gael.
Cllr O’Connell said Fianna Fáil appear to be edging away in the opinion polls and this is causing unrest.
He said the question of the party’s leadership is now coming to the fore.