The 30-year-old midfielder joined the Hammers from United Arab Emirates outfit Al Jazira in the summer of 2015 on an initial loan deal before making his move permanent the following year.
Lanzini, regarded as a fan favourite at the London Stadium, scored 32 goals and registered 28 assists in 226 appearances for West Ham across all competitions during his eight-year spell with the club.
The Argentine won the club's Goal of the Season award twice for his last-gasp long-range equaliser in a 3-3 draw at Tottenham Hotspur in 2020-21 and for his stunning volley against Crystal Palace in 2021-22.
Lanzini made nine appearances in the Hammers' triumphant Europa Conference League campaign in 2022-23 and scored in both home and away win over Danish club Silkeborg during the group stage.
The midfielder, who has been linked with a return to his homeland with River Plate, marked his final home game for West Ham with a goal as a last substitute in the 3-1 Premier League win over Leeds United at the London Stadium in May.
Manager David Moyes has paid tribute to Lanzini and his contribution to the club, telling the club's official website: "On behalf of the players and coaching staff, I would like to thank Manu and wish him all the very best for the future.
"Manu has been a tremendous servant for the football club and a brilliant player for me in my time here as manager. In my first spell he played a key role in helping us to stay up, and he has always been someone I can rely on.
"He has some wonderful memories at West Ham - including that famous equaliser at Tottenham when we came back from 3-0 down to draw 3-3. People still talk about that goal and what that result did to increase the confidence and belief around the club at that time.
"Manu is a fantastic professional, a really good lad, and whichever club he joins will have an excellent player on their hands.
"We are sorry to see him go, but he deserves the opportunity to play more regularly elsewhere, and we wish him well in his next adventure."
West Ham have also confirmed that development squad players Ademipo Odubeko, Armstrong Oko-Flex and Brian Kinnear, as well as scholar Asher Falase, will all be leaving the club this summer, while 19-year-old defender Jamal Baptiste is also due to depart when his contract expire on June 30, but he has been offered a new contract.
Experienced defender Angelo Ogbonna has also been offered a new deal and the 35-year-old Italian is set to extend his eight-year stay at the London Stadium, subject to personal terms being agreed.
Ogbonna missed the majority of last season with a serious ACL injury but he recovered in time for the 2022-23 campaign and went on to feature 31 times across all tournaments, including 13 Premier League starts and eight outings in the Europa Conference League.
Meanwhile, Moyes is expected to remain in charge after holding talks with chairman David Sullivan last week, but club captain Declan Rice is set to leave, with Arsenal, Manchester City, Manchester United and Chelsea all expected to battle for the signature of the £120m-rated midfielder.