In return, Toronto sent the Predators defenseman Brett Lebda and forward Robert Slaney.
The deal also includes a conditional fourth-round pick that Nashville acquires if Lombardi plays more than 60 games over the next two seasons.
If he doesn't, Toronto takes the Predators pick in 2013.
The 29-year-old Lombardi missed all but two games of last season because of a concussion suffered in October.
The forward has 89 goals and 147 assists in 446 games with Calgary, Phoenix and Nashville.
Franson had career highs last season with eight goals and 21 assists. He has 14 goals and 36 assists in 141 career games with Nashville.