The Nationals also were to get a player to be named as part of the deal and the Blue Jays will receive cash considerations.
Revere batted a combined .306 with two home runs, 45 RBIs and 31 stolen bases last season with the Philadelphia Phillies and the Blue Jays, where he landed at the trade deadline in July in a deal for two minor league pitchers.
Storen, 28, wanted out of Washington after losing the closer role to Jonathan Papelbon, who was acquired from the Phillies last year. Storen moved into a setup role and did not fare well before his season ended Sept. 9 when he broke a thumb after slamming his locker in frustration.
Storen appeared in 58 games in 2015 and was 2-2 with 29 saves and a 3.44 ERA in 55 innings.
It's unclear whether the Blue Jays will use Storen as their closer next season or stick with Roberto Osuna.