Griezmann scored 133 goals in 256 appearances for Atletico Madrid after arriving from Real Sociedad in 2014.
France international Griezmann signed a five-year deal with Atletico in June 2018 with a buyout clause of €200 million ($225m) which dropped to €120 million (135m) at the start of this month.
The move comes after weeks of speculation following Griezmann’s announcement that he intended to leave Atletico.
In the video aired in May on Atletico’s social media channels, the World Cup winner said: “I wanted to tell you that I’ve made the decision to leave to experience new things and have new challenges. It’s been very hard for me to take this path, but I feel that it’s what I need.
“I wanted to thank you for all the love you’ve given me in these five years, in which I’ve won my first important trophies with a club. They were incredible moments that I will always remember. The truth is you’re in my heart.”
He added: “It isn’t easy for a player to receive so much love. That’s why I wanted to tell you today so that you’re the first to know after the [head] coach and the club.
“I’ve given my all on the pitch, I’ve tried to behave well. Thank you so much, and goodbye.”