Half of people think Tony Blair should quit sooner rather than later, a BBC Newsnight poll suggests.
The survey suggests Iraq has hurt Mr Blair's reputation
The prime minister has said he will step down before the next election.
But 29% of those asked said he should resign now and another 21% said he should go within a year.
The poll also showed some solid backing for Mr Blair, with 23% saying he should stay longer than a year before standing down and 21% saying he should change his mind and contest the next election.
The survey, to be unveiled on BBC2's Newsnight programme on Monday evening, suggests that the Iraq war has damaged Mr Blair's popularity.
Some 52% said their opinion of him had gone down as a result, while 5% said their opinion had gone up.
However, 63% of the 1,006 adults questioned, said their opinion of Gordon Brown remained the same following the war.
Amid warnings that Iraq may be moving towards civil war three years after the invasion, some 60% said military action was the wrong thing to do with fewer than a third, 31%, believing it was the right thing to do.
Of those asked, 40% said British and American troops should pull out of Iraq "now", 16% within six months and 37% that they should stay as long as necessary.
* ICM interviewed a random sample of 1,006 adults aged 18 and over during the weekend 17th and 19th March 2006. Interviews were conducted by telephone throughout the country and the results have been weighted to be representative of all adults.