Johnson, 29, will sign a two-year contract worth about $2 million annually.
Before free agency started, Tampa Bay officials expressed interest in retaining either Johnson or Shaun King, but the Bucs lost Johnson on Sunday and could lose King to Chicago, who is expected to put a full-court press on King for his services.
Johnson signed a one-year deal with Tampa Bay last offseason, but he didn't quite live up to head coach Jon Gruden's expectations. Johnson spent most of last season as the Bucs' second-string quarterback and even had a brief stint as the team's third-string quarterback.
Johnson completed 57-of-88 passes (64.8 percent) of his passes for 536 yards and he threw one touchdown and two interceptions. Johnson was 2-0 as a starter for the Bucs, but he failed to lead the offense into the end zone in both of those contests and he was sacked 19 times in a total of six appearances.
Johnson will go into training camp as a backup to Redskins second-year QB Patrick Ramsey. He will be reunited with Washington offensive coordinator Hue Jackson, who was a member of the University of Southern California coaching staff when Johnson attended (1991-94).
Brad Johnson and Greg Zolman are the only two quarterbacks currently under contract with the Buccaneers.