In return, Toronto sent the Predators defenceman 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.
Lombardi, 29, missed all but two games of last season due to a concussion suffered in October.
The five-foot-11, 195-pound forward has 89 goals and 147 assists in 446 games with Calgary, Phoenix and Nashville.
Franson, 23, had career highs last season on the Predators' blue-line with eight goals and 21 assists.
The six-foot-five, 213-pound Franson has 14 goals and 36 assists in 141 career games with Nashville.
Lebda, 29, goes to the Predators after registering just one goal and three assists in 41 games with Toronto last season.
The five-foot-nine, 194-pound defenceman has 19 goals and 53 assists in 367 games with Detroit and Toronto.
Slaney, 22, has yet to play a game in the NHL.
The six-foot-two, 203-pound Slaney spent the majority of last season with the ECHL's Reading Royals, registering seven goals and 17 assists in 61 games.