Diego Simeone’s Spanish side will pay an initial £64.4m (75m euros) with a further £17.1m (20m euros) in potential add-ons for the Argentina striker.
The 24-year-old’s departure is a record sale for City, eclipsing the £50m they received from Chelsea for Raheem Sterling in 2022.
They paid Argentine side River Plate £14.1m for Alvarez in January 2022.
Alvarez said City – where he won six major trophies during his two-year stay – will always “have a special place” in his heart.
“Today I say goodbye to this amazing club, with a lot of emotion,” he said.
“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.”
He won the treble during his debut season at the club, as well as the 2022 World Cup midway through the season with Argentina.
The striker departs Manchester having scored 36 goals in 106 appearances for City.
City director of football Txiki Begiristain said Alvarez has been “a pleasure to work with”.
“The trophies he has won already tell their own story – he is a winner,” added Begiristain
“I look forward to watching him develop further in the remainder of his career.”
City start their Premier League campaign with a trip to Chelsea on Sunday.
Atletico Madrid visit Villarreal on 19 August on the opening weekend of the La Liga season.
