Some 16 per cent want the Prime Minister to go now, with a further 23 per cent saying he should leave office before the next election. And 21 per cent want him to stand down after the election. However, 35 per cent of members want him to fight the election after next.
The Guardian/ICM poll showed 55 per cent of members would back Mr Blair in a leadership campaign, while just 30 per cent would support Gordon Brown.
The results come after Mr Blair made clear in a weekend newspaper interview that he intended to lead Labour into the next General Election.
However, the Lord Chancellor, Lord Falconer of Thoroton, called on Mr Blair to serve out a full third term in office if he was elected again.
The poll also showed the Government’s support fell three points to 36 per cent, with the Tories remaining at 34 per cent.
The Conservatives said the poll showed Mr Blair had lost the respect of his party.
A spokesman said: "The way the Labour party machine has conducted itself over the last seven years has fuelled cynicism and a lack of trust in government and people are looking for a fresh alternative."
• A House of Commons committee is to investigate the Government’s use of inquiries to deal with controversial issues.
The all-party group wants to look at whether the current system is adequate for the job.