A significant majority of voters believe the Taoiseach Brian Cowen should not lead the Government into the next election.
The Irish Times latest opinion poll also suggests Brian Lenihan is the favourite to take over.
Earlier this week 1,000 voters across the country were surveyed by IPSOS MRBI for the newspaper.
Asked if the Taoiseach should step down before the next election, almost two thirds, 61%, said he should.
While 10% didn't know, 29% felt Brian Cowen should stay on and lead the Government into an election.
When asked who his successor as leader of Fianna Fáil should be, most went for the Finance Minister.
Up 39% went for Brian Lenihan, 18% think it should be Micheál Martin, 8% opt for Mary Hanafin, 6% for Dermot Ahern, 5% for others and one in four voters, 24%, don't know.