Gwyneth Dunwoody served as the area's MP for 34 years
The first opinion poll ahead of the Crewe and Nantwich by-election, to be held later this month, puts the Tories four points ahead of Labour.
The ICM survey for the Mail on Sunday suggests the Conservatives could win their first by-election in a seat held by another party since 1982.
It puts the Tories on 43%, Labour on 39% and the Liberal Democrats on 16%.
The by-election was triggered by the death of Gwyneth Dunwoody last month. She was the longest-serving female MP.
The poll suggests a 10% swing since the last general election which would turn Labour's 7,078 majority in the seat into a Conservative one of more than 1,000.
It comes after what Gordon Brown described as a "disappointing" result for Labour in local elections held earlier this month.
Tamsin Dunwoody, 49, aims to defend her mother's Labour majority on 22 May.
Edward Timpson will fight the seat for the Conservatives, while Elizabeth Shenton will stand for the Liberal Democrats.