A day after Cameron and Tyler Winklevoss indicated they wouldn't take their fight with Facebook Inc. founder Mark Zuckerberg to the Supreme Court, they pushed ahead with another suit.

In a status report filed on Thursday in U.S. District Court for the District of Massachusetts, the twins and their business partner Divya Narendra told the court that they would push for discovery in a claim that Facebook "intentionally or inadvertently suppressed evidence" during 2008 settlement proceedings over whether Mr. Zuckerberg stole...