The case has been raised by four of his constituents in Orkney and Shetland.
They argue he misled the electorate over a memo claiming SNP leader Nicola Sturgeon would rather have seen David Cameron become prime minister.
The case is due in court next month with more than Â£60,000 already raised but now more is being sought.
Mr Carmichael had initially denied leaking the confidential memo to the Daily Telegraph.
He said that the first he had heard of it was when he received a phone call from a journalist.
The MP later admitted full responsibility for sanctioning its release, and accepted that the "details of the account are not correct".
The official cabinet office inquiry into the leaking of the memo found that Mr Carmichael had given permission to his former special adviser Euan Roddin to hand details of the memo to the newspaper.
Campaigners raised Â£60,000 in crowd-funding to lodge the legal challenge to his election under the Representation of the People Act 1983.
The team behind the campaign to overturn the election of Liberal Democrat MP Alistair Carmichael has launched a second appeal for funding.