The latest opinion poll suggests that a majority of people believe Mary Harney should step down as the Minister for Health in light of the winding down of the Progressive Democrats.
In today's Irish Times TNS/mrbi poll, 63% of respondents said Ms Harney should not remain as Minister.
However, 30% said she should remain and 7% did not know.
Meanwhile, 50% said former Taoiseach Bertie Ahern would make a better leader in dealing with the current economic difficulties, compared with 24% who favoured Brian Cowen.
26% of those surveyed said they did not know.
Minister for Transport Noel Dempsey,has said the results of recent opinion polls were very disappointing.
Mr Dempsey said the question on Bertie Ahern was a hypothetical one, and that Brian Cowen had the capacity to focus on stabilising the economy.
Chief Whip Pat Carey has said the Minister for Health has the full confidence of the Taoiseach and the Government and should stay in her position.
Flanagan critical of poll question
Meanwhile, Fine Gael's Charlie Flanagan criticised a question in the poll asking whether party leader Enda Kenny or Richard Bruton would be the better leader in the economic downturn.
The poll showed 46% of voters backed Mr Bruton, 28% supported Mr Kenny and 26% did not know.
Mr Flanagan said there was no question of Mr Kenny not leading the party into the next election.