Dawson, the second-leading scorer in team history behind Hall of Famer Lou "The Toe" Groza, is in the final year of his deal. He has been asking for a new contract for several years. Dawson also sat out Cleveland's offseason training activities last season.
Browns general manager Tom Heckert would not say if Dawson's absence Monday was because of a contract dispute.
"I really don't want to get into it, but that's not the whole thing with him," Heckert said before speaking at a Browns Backers banquet.
Dawson has been with the team since its expansion return in 1999. He missed five games last season with a calf injury, but has been one of Cleveland's steadiest players throughout his career.
The Browns began their second season of organized team activities under coach Eric Mangini without Dawson and five restricted free agents: running back Jerome Harrison, fullback Lawrence Vickers, linebacker Matt Roth, safety Abram Elam and linebacker D'Qwell Jackson.
Heckert said the team has no plans to trade any of the free agents. Roth, claimed off waivers from Miami late last season, asked the Browns to trade him.
"It's not going to happen," said Heckert, adding the club had been in contact with Roth's agent, Drew Rosenhaus.
Heckert said none of the free agents, each of whom was offered a one-year tender, has any leverage. He also pointed out that it is difficult for him to make long-term decisions on players he doesn't know well. Heckert and president Mike Holmgren didn't join the team until after last season.
"Eric [Mangini] has only been around them for a year and we haven't seen them at all," Heckert said. "It's tough for us to make a long-term commitment, but it's hard when you haven't been around them. That's not saying they aren't good players. It's tough to do when you're talking financially about quite a bit of money and we don't feel comfortable doing it right now."