Dye did not say how much money was offered to him, and Nationals general manager Mike Rizzo declined comment on whether the team offered the right-handed-hitting slugger a contract.
The team had interest in Dye since it released right fielder Elijah Dukes in late March. One source close to the Nationals felt $4 million was a fair price for Dye, who made $11 million last year with the White Sox. Washington is platooning Willie Harris and Willy Taveras in right field.
"I'm sure it was as much as they could do," said Dye about Washington's offer. "With the economy, they can only offer so much. I'm sure with that team and that payroll, what they had to offer was their best. I look at the whole situation: Living in Phoenix, Arizona -- this will be my [15th] year in the big leagues -- and Washington being on the East Coast, that's not like one of the first teams that is going to jump out at me.
"I'm not going to take my family all the way to Washington. With the offer that they made, it would maybe have to be a little bit more than other clubs have offered me in the past. I'm just not going to take my family all the way to Washington just to play.
Dye, 35, is coming off a season in which he hit .250 with 27 home runs and 81 RBIs. Dye said he has been offered deals in the $1 million range and some teams have indicated he could be a fourth outfielder.
'When you get offers from a lot of teams that are in the $1 million range, that's a negative in itself, no matter what," Dye said. "I definitely want to play. I'm working out, staying in shape and just wait for the right situation. [My family and I] are OK with it."