A majority of the electorate want a general election before the end of the year, according to the second part of an opinion poll due to be published in tomorrow's Irish Times.
However, the Ipsos MRBI poll found no clear choice for the next Taoiseach.
The poll, conducted on Tuesday and Wednesday among 1,000 voters around the country, asked participants to choose their preference for Taoiseach after the next election, between the leaders of Fianna Fáil and Fine Gael.
21% said they would like to see Brian Cowen back in the Taoiseach's office after the election.
30% opted for Fine Gael leader Enda Kenny, but 40% said they did not like either of those choices while 9% said they did not know.
Presumably, quite a few voters would have chosen Eamon Gilmore if his name was on the questionnaire - it probably will be the next time, given the party support figures.
Asked if they would like an election this year, 57% said they would; 35% did not want an election in 2010; and 8% said they did not know.
Questioned about the three pending bye-elections, 60% said they should be held now, 24% did not want them held immediately, and 16% had no opinion.
Speaking in Ennis earlier this evening, the Taoiseach dismissed suggestions that the poll undermined his position as Taoiseach and leader of Fianna Fáil, particularly coming into the confidence motion in the Dáil next week which he said the Government would win.
Mr Cowen said he would not walk away from Government and would make what he called the tough decisions on the economy.
'I'm going to make sure that I see us through this period; that we take the decisions we have to take and that we face up to the problems,' he said.
He made the comments after a poll in today’s Irish Times showed Labour had become the most popular political party in the country for the first time.
Earlier, a prominent Fianna Fáil backbencher said the leadership of the party must be discussed after the opinion poll result.
Cork North-Central's Noel O'Flynn said the figures would signal an electoral ‘wipe-out’, and that the party leader must look at the situation.
He said it would be wrong if the parliamentary party did not discuss the leadership and whether there should be a change.
However, he pointed out that there could not be a change unless someone came forward to say they wanted to take over the leadership - and so far, nobody had.
Labour leader Eamon Gilmore described the poll as a 'historic first', which reflected people's desire for change.
Mr Gilmore said it showed that a Labour-led Government was a possibility after the next election.
Asked who his preference for Tánaiste was in such a government, Mr Gilmore replied 'Joan Burton'.
Asked about Enda Kenny's poor showing, he said the poll was not about individuals, it was about party support, and the increase in support for Labour was the significant change.
He ruled out accepting support from Fianna Fáil to form a government, saying it was necessary for the country's recovery that Fianna Fáil be moved from office.
The Ipsos MRBI poll shows a slump in support for Fianna Fáil and Fine Gael, and for their leaders.
Carried out on Tuesday and Wednesday, the poll shows Fianna Fáil down five points since January at just 17%, while Fine Gael is also down, by four to 28%. Labour is up eight points to 32%.
The Green Party is unchanged at 3%, Sinn Féin gains 1% to 9%, and Independents are unchanged at 11%.
Eamon Gilmore also has the highest satisfaction rating of 46%, while satisfaction with Brian Cowen is down 8% to 18%, while Enda Kenny drops 7% to 24%.
Satisfaction with the Government is also down since January, by 7% to 12%, while 83% say they're dissatisfied.