Kilkenny Fine Gael Councillor David Fitzgerald believes a split in the party is possible if Enda Kennedy stays on as leader.
He was reacting to the news that his party colleagues have called for the Taoiseach to set out a timeline for his departure.
Senior government ministers have been defending the Taoiseach, saying any challenge to his leadership is a mistake.
There are three front runners on the list to replace Enda Kenny – they are Simon Coveney, Leo Varadkar and Frances Fitzgerald.
Speaking to KCLR this morning, Councillor Fitzgerald says he would be backing Simon Coveney, despite spending time with all three of them.