The deal is understood to be worth over $124 million with a further $33 million in potential add-ons.
It represents a significant profit on a player signed from River Plate for $23 million and underlines City manager Pep Guardiola's willingness to let unhappy players leave.
He has won the Premier League twice and the Champions League, FA Cup, Super Cup and Club World Cup all once. He also won the 2022 World Cup with Argentina and the Copa America earlier this year.
Alvarez scored 36 goals In 103 appearances in all competitions for City, but despite being first choice for Argentina, he was behind Erling Haaland in the City pecking order.
The 24-year-old told City's website: "Today I say goodbye to this amazing club, with a lot of emotion. These have been two very special years. During this time, I have grown and learnt a lot - both as a player and as a person."
City director of football Txiki Begiristain said Alvarez's "attitude has been first-class since he first walked through the door here.
"That's why he has improved so much," Begiristain added. "The trophies he has won already tell their own story - he is a winner. I look forward to watching him develop further in the remainder of his career."
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