Chile international Sanchez has played 142 matches for Inter across two spells, winning two Serie A titles, two Italian Super Cups, and one Italian Cup with the team.
Cuadrado arrived as a free agent from Juventus before last season started and played 12 matches for the club.
The Dutch international Klaassen arrived from Ajax Amsterdam as a free agent and signed a one-year deal that was not extended.
Manager Simone Inzaghi also released midfielder Stefano Sensi and sent goalkeeper Emil Audero back to Sampdoria after a season on loan as backup to Yann Sommer.