FanDuel Inc. on Friday said it is no longer accepting new deposits from players in New York, though it will continue to operate its full slate of contests this weekend.

The daily fantasy-sports company and its chief rival, DraftKings Inc., earlier Friday filed separate lawsuits seeking to stop New York’s Attorney General Eric Schneiderman from banning fantasy-sports games in the state. DraftKings didn’t immediately respond to questions about whether it, too, was halting new deposits.

FanDuel...