The club's owners, current manager Mark Hughes and former boss Tony Pulis were among the names he thanked.
"I would like to say thank you to everyone at Stoke Football Club, the chairman and Coates family, Mark Hughes and Tony Pulis, goalkeeper coach Andy Quy, all my team mates and especially the fans who have made me and my family so welcome," Begovic wrote in a post on Twitter.
"I have loved my time at Stoke, but feel I had to take this opportunity for me to continue to develop and challenge myself as a player.
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