The poll by Mori put Labour on 47 (down 2) the Conservatives on 26 (down 2) and the Liberal Democrats on 23 (up 3).
The findings from a survey for the Times of 1,917 voters support evidence from Labour and Liberal Democrat canvassers in the local elections, who have reported a collapse in Conservative support in some areas.
Leaders of both parties believe the fall in support for the Conservatives could lead to the Liberal Democrats seizing seats next week in Labour and Tory heartlands.
Labour believes that Oldham is one of the areas in the north-west of England where the Liberal Democrats will be helped by the Conservative collapse.
An NOP poll published in the Independent on Saturday gave Labour 48 (down 2), the Conservatives 26 (no change), and Liberal Democrats 21 (up 1).