The Recording Industry Association of America filed a request with Judge Marilyn Hall Patel asking for a summary judgment, which would effectively shut down the company's internet music service.
The RIAA wants an October 1 hearing on the summary judgment request. No date has been set for a full-blown trial in the case.
In its court filing, the RIAA insists that Napster "knowingly and willfully" set out to build a business based on copyright infringement on an unprecedented scale.
Judge Patel slapped an injunction on Napster after ruling last year that the company's free swapping of copyrighted songs is illegal. Record companies are seeking the maximum damages of $100,000 (£70,427) for each work infringed and the total amount of damages could exceed $100m (£70.4m)
In February, Napster offered $1bn (£704.3m) to settle the case, based on the potential earnings of a subscription-based service. The RIAA rejected the offer.
The company has promised to develop a legal subscription service later this year. It emerged yesterday that the company plans to charge users $5 (£3.52) a month for the service.