Julian Alvarez thanked Manchester City for "two very special years" after completing his move to Atletico Madrid.
The deal is understood to be worth £81.5million in total, with an initial fee of 75m euros and 20m in add-ons amounting to a record sale for City. Alvarez has signed a six-year contract.
It represents a significant profit on a player signed from River Plate for £14m and underlines manager Pep Guardiola's willingness to let unhappy players leave.
Alvarez enjoyed astonishing success in his debut 2022-23 season for City. By Christmas of that campaign he was a world champion with Argentina and went on to win the treble of Premier League, FA Cup and Champions League with his new club, who retained the English top-flight title last term.
Alvarez 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.
"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 team-mates, 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.
"To the fans, thank you from the bottom of my heart for the great support and for making me feel at home from the very first day. I will always be extremely proud to have been part of this family.
"Manchester City will always have a special place in my heart. I wish you every success in the future and I will keep on supporting the club wherever I may be."
Speaking ahead of Saturday's Community Shield win over Manchester United, Guardiola said he hoped Alvarez would find "happiness" in Spain.
"He played a lot last season and in some moments with Erling (Haaland) but I understand in some important moments he thought he was going to play," Guardiola said.
"He said he wants a new challenge and it is what he has to do. I don't like to tell players what they have to do. Contracts are nice but they can be broken. If they are not happy, why should they be there?
"If they think happiness is around the corner, go and take a look. Both clubs are happy with the agreement, I look forward to see him back to say bye to all of us. It was a joy to work with him, I learnt a lot from him and hopefully he can find what he was looking for."
