Tinordi's contract will be worth $700,000 at the NHL level and $300,000 at the American Hockey League level each of the next two seasons.
The 27-year-old Tinordi had eight goals and 14 assists this season in 75 games with the Milwaukee Admirals, the Predators' AHL affiliate.
Tinordi, who is 6-foot-6, has six assists in 53 career NHL games since the Montreal Canadiens selected him in the first round of the 2010 draft.
Tinordi spent parts of four seasons with the Canadiens before getting traded to Arizona. He hasn't played in the NHL since appearing in seven games with the Arizona Coyotes in 2015-16.
He has 20 goals and 65 assists in 365 career AHL games.