The Argentine joined the Hammers from Emirati club Al Jazira in 2015, at first on loan before signing permanently a year later. He has scored 32 goals in 226 appearances.
He has been linked with a return to his former club River Plate and will leave West Ham upon the expiry of his contract.
David Moyes said: “Manu is a fantastic professional, a really good lad, and whichever club he joins will have an excellent player on their hands.”
The club’s sporting director, and Lanzini’s former Hammers team-mate, Mark Noble said: “Manu is what I would call a player’s player. Until you have trained and played with him, you really don’t appreciate just how gifted and talented he is. Technically superb, fantastic vision, and so unselfish in how he uses the ball and helps the team.
“For the last eight years, he has given absolutely everything for the club. It was a true pleasure to play alongside him in West Ham’s midfield, and to count him as a great friend during his time here.”
Meanwhile, Angelo Ogbonna has been offered a new contract by the Europa Conference League champions, while Ademipo Odubeko, Armstrong Oko-Flex, Brian Kinnear, Asher Falase and Jamal Baptiste will be released. Baptiste, 19, had been offered a new contract but will leave the club.
Scholars Sean Tarima, Ryan Battrum, Billy Bates and Mason Terry have all signed their first professional contracts with West Ham.
