The National Football League has asked a federal judge to appoint a special investigator to probe what it describes as a widespread fraud that has resulted in hundreds of millions of dollars in false claims to a fund meant to compensate former players for head injuries.

The motion alleges fraudulent schemes by doctors, lawyers and players to illicitly tap the uncapped fund, which is potentially valued around $1 billion. The league established the fund as part of a settlement agreement with players, and the NFL has so far funded...