There is speculation that the Tory candidate for Eastleigh will be Maria Hutchings.
Ms Hutchings who fought the seat in 2010.
Her chances of achieving the necessary 3.6% swing to win have been boosted by UK Independence Party leader Nigel Farage's decision not to stand.
It comes after Deputy Prime Mininister Nick Clegg today launched the race for the seat, vacated by Chris Huhne on Monday.
The date for the election has been set for Feburary 28.
Labour sources are hoping for a strong showing, but privately accept they have little chances of winning.
Shadow chief Treasury secretary Rachel Reeves dismissed Mr Clegg's attempts to distance his party from the Tories.
''The Liberal Democrats will be judged on what they do not what they say, and their record is the Tory record - one of economic failure,'' she said.
''Nick Clegg failed to deliver a mansion tax, but went along with a £3 billion tax cut for the richest. And his cuts to tax credits will dwarf any gain from the rise in personal allowance for millions of working families.''
Simon Hughes, deputy leader of the Lib Dems, said the party will be ''all guns blazing'' at the next by-election.
He said: ''We don't take anything for granted, we have a very well run council, we run the council, and a fantastic reputation for work.
''So we're going to go in all guns blazing and this weekend pick a candidate, who will be up and running as of Saturday.
''I think people increasingly do know that it's us who've delivered the tax cuts.
''It's the most important thing you can do, to put money back in the hands and the pockets of families who are struggling to pay the bills.''
Asked whether Chris Huhne had damaged the image of the party, he said: ''Of course it is a very unsatisfactory state of affairs, both for him and his family, and for the party. He's resigned as MP, we fight a by-election.
''Chris Huhne paid the penalty, we move on.''