Caballero only moved to City earlier this month.
He told a press conference on Sunday: "Manchester City is a big opportunity in my sporting career and it would have been impossible to reject this chance. I've taken an important step for myself and my family.
"I have no more words than to say thank you to a club that allowed all this to happen for me.
"I'm leaving a club where I've been very happy, because I was treated phenomenally well, like I've never experienced anywhere else and this affection is something I'll take with me wherever I go.
"I always tried to give everything I could for this team and these colours.
"Hopefully the door will remain open for me to return one day. A part of my heart will always stay here and the other part is going to England.
"Now all I would like to do is wish the very best of luck to all my former team-mates. I'm really going to miss them all and I'll keep trying to grow as both a player and a person."
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