The 28-year-old Griezmann has a contract until 2023 with Atletico, but has a buy out clause of 120million euros ($134million).
He is reportedly the target of La Liga rivals Barcelona.
“Antoine Griezmann has told the club he will not continue as a Rojiblanco (red and white, the club’s colours) next season,” the club said on Twitter.
Within a few moments the player himself then posted a video explaining himself.
He said: “It’s been an incredible five years. Thank you for everything,” he said.
“I wanted to tell the fans who have always given me a lot of love that I have taken the decision to leave, to see other things, to have other challenges.
“The truth is it has been difficult to take this route but it is what I feel I need and I would like to thank all of you for the love you have shown me during these five years.”
While in Atletico, he scored 133 goals in 256 appearances, since joining from fellow Spanish outfit Real Sociedad in 2014.
He won the Europa League, the Spanish Super Cup and the Uefa Supercup during his time at Atletico and has been the team’s top scorer in each of the last five seasons.