Most voters believe Tony Blair should resign as prime minister within a year, according to a News of the World poll.
Mr Blair has seen his popularity plummet in recent months
Of those polled, 42% want him to step down now and a further 15% within a year. Some 21% said they wanted him to stay until after the general election.
Mr Blair has pledged not to lead Labour into another election but he said he expects to serve "a full third term".
A third of the 1,012 UK adults surveyed think neither Mr Blair nor Chancellor Gordon Brown should run the country.
Blair or Brown?
Mr Blair's pledge over not leading Labour into the next election has prompted speculation about the exact timing of his departure amid crises over party funding and education reforms.
The poll, conducted by ICM on behalf of the News of the World, points to a big drop in the prime minister's popularity over the last few months.
In November only 28% wanted Mr Blair to step down straight away and a quarter preferred him to carry on beyond the next election.
And, in less than two years, he has also lost a nine-point lead over the chancellor over the issue of who would make the best prime minister.
The pair are now level at 30% - down from 45% for Mr Blair and 34% for Mr Brown in June 2004.
However, the number of voters who believe neither of the men make a good prime minister has more than trebled in that time from 9% to 31%.
The uncertainty over when and how the leadership would change hands has prompted 15% of the respondents to say they were less likely to vote Labour.
Just 4% said they were encouraged to back the party because of it.
However, the vast majority of respondents - 78% - said the situation would make no difference at all.
A random sample of adults were interviewed across the UK by telephone on 29 and 30 March and the results were weighted to represent the population as a whole.