Navarro, one of the best point guards in Europe, was originally drafted by the Wizards in 2002 with the 40th overall pick in the second round.
The 27-year-old never appeared in Washington's team, opting instead to play for Barcelona.
But the Spanish club agreed last month to a buyout that would allow Navarro to move to the NBA.
"We're pleased to acquire a first-round pick," said Wizards president Ernie Grunfeld in a statement.
"By trading the rights to Juan Carlos Navarro...we have greater flexibility to build our team."
Washington was 41-41 last season and lost in the first round of the playoffs to the Cleveland Cavaliers. Memphis finished 22-60, the worst record in the NBA.
Navarro will play alongside fellow Spaniard, forward Pau Gasol, in Memphis.