Johnson, a fourth-year pro, was drafted by Tampa Bay in the fifth round of the 2008 draft by now-Redskins general manager Bruce Allen. He has started five games in his career.
Johnson is reunited with Jim Harbaugh, who coached him at the University of San Diego, and is expected to back up Alex Smith, whom the team re-signed this week.
Washington is expected take Baylor's Robert Griffin III with the second pick in the NFL draft, and the team last Saturday agreed to terms to retain Rex Grossman. The team also has John Beck and Jonathan Crompton under contract.