The NFL’s locked out players are asking a federal judge for $640 million in damages in their dispute with the league over television contracts.

Judge David S. Doty also could award the players punitive damages that could push the total to around $2 billion. Doty did not issue a ruling after a hearing Thursday in Minneapolis on penalties and remedies for the way the NFL negotiated $4 billion in annual national television contracts.

Doty has previously ruled that the NFL improperly left money on the table when it negotiated the TV contracts. The players contend the owners effectively set up a lockout fund in a deal that called for the television networks to pay them even in the event of a work stoppage.

The $640 million represents the amount players consider their portion of the lost revenue from TV and other related sources. In addition, they could be awarded as much as $1.92 billion in punitive damages.

The league makes about $4 billion annually in TV rights fees. The players have asked Doty to deny the league access to those funds if the lockout continues into the season.

The league is asking Doty to give the players only the $6.9 million previously awarded to them by Stephen B. Burbank, the sport’s special master, who decided the case before the players appealed to Doty.

Doty ruled in March that the structure of the league’s TV contracts violated a settlement agreement between the NFL and the players that formed the basis of their longtime collective bargaining agreement. Doty oversaw that now-expired labor deal, and expressed displeasure Thursday about the current standoff between owners and players.

“I didn’t think we would have this hearing,” Doty said, according to the Associated Press,“and I’m a bit disappointed we are having it.”

The players have been locked out since March 12, except for several days last month when a federal judge ordered the end of the shutdown. An appellate court later temporarily reinstated the lockout. The two sides are awaiting a June 3 hearing before the U.S. Court of Appeals for the Eighth Circuit in St. Louis on the order that lifted the lockout.