The survey, conducted immediately after the resignation of disgraced former MP Chris Huhne, put the Tories on 34 per cent, with Lib Dems on 31 per cent, Labour on 19 per cent and Ukip fourth with 13 per cent.
But Lord Ashcroft warned the Tories cannot expect to gain much of a boost in the February 28 by-election from the circumstances of Huhne's fall from grace, as the poll suggested that few voters were likely to ''switch out of disgust''.
The figures represent a fall of almost 16 points in Lib Dem support since the 2010 general election, when Huhne scooped 46.5 per cent of the vote. Tories are down five points since the election and Labour and Ukip both up nine.
Lord Ashcroft, a major Tory donor who was deputy chairman of the party until 2010, said the poll suggested both coalition parties have ''everything to play for''.
''There is some more encouraging news for the Tories,'' he added.
''Conservative voters are the most likely to be sure how they will vote, and Lib Dems who say they may change their minds are more likely to switch to the Tories than any other party.
''However, Labour voters are more likely to be open to changing, and are more likely to move to the Lib Dems if they do.
''Only a fifth of Eastleigh voters would rather see Ed Miliband as Prime Minister than David Cameron, and the Conservatives lead on getting the economy growing and creating jobs, as well as dealing with the deficit.''
But there were indications that Nick Clegg's party may gain votes through their performance locally.
The Deputy Prime Minister yesterday said Huhne had been a good constituency MP, and two-thirds of those taking part in the poll agreed that ''the Lib Dems do a good job locally in my area''.
''Winning a seat from the Liberal Democrats is much easier when the incumbent is out of the picture,'' said Lord Ashcroft.
''But Eastleigh has been in Lib Dem hands for a long time, and the party has a strong local government base in the borough. Huhne was a popular and, by all accounts, assiduous MP, and many in Eastleigh will be sad to see him go and circumstances of his departure. The Tories should not expect many to switch out of disgust.
''So the Conservatives, with an initial lead and reasonable support for their national performance, versus the Liberal Democrats on home territory. This is going to be bruising - but fascinating.''
- Some 1,006 adults were interviewed by telephone in the Eastleigh constituency on February 4 and 5.