The Iceland international is also being linked with English Premier League clubs Tottenham (where he was on loan last season), Fulham and West Ham, but Wizards president Robb Heineman confirmed the club had made a bid for his services.
"We've made a formal offer to Icelandic striker Eidur Gudjohnsen," Heineman told Big Soccer.
"He's recently been at Tottenham, and I'm relatively certain they will make an offer as well.
"This is the type of guy we want to go after: A world-class player who knows how to win championships.
"I'm hopeful we get this done, but it's going to be a bit of a waiting game."