The two, aged 22 and 27, were found guilty in April and given suspended six-month sentences and fined €5,000 ($6,900) each for operating the site, Undeadlink, which provided links to illegal copies of copyrighted material.
The request for damages is expected to be considered at a civil hearing in the eastern town of Bourg-en-Bresse, but the decision may be deferred to another date.
Among those demanding damages are French film distributors and US studios Columbia, Disney, Paramount, Tristar, 20th Century Fox, Universal and Warner Bros.
They are claiming €2.5 million ($3.44 m) in damages and interest, alleging that the site provided access to 520,000 links and facilitated around 80 million downloads, earning the pair around €24,000 ($33,000) between May and December 2011.
SACEM, a French royalty collection body, and SCPP, a group protecting the rights of record and video producers, are separately seeking €9,887 ($13,619) and
€201,872 ($278,078) in damages, respectively.