The Argentine has left City for Atletico Madrid in a big money deal, with the fee possibly rising to £82 million.
City have snagged a club record sale, but will now have to contend with finding a replacement.
On City, Alvarez said: “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.
“To the directors and the technical staff, thank you for your confidence and support. Your teachings and vision have helped me to develop so much.
“To my teammates, thank you for every moment we have shared, for the effort and dedication in every training session and every match. I have learnt a lot from each and every one of you and I remain with memories and friendships that will last forever."