Griezmann is among the favourites to win the Ballon d'Or after helping Atletico and France reach the Champions League and Euro 2016 finals respectively last season.
However, compatriot Ribery told Sport Bild that "Griezmann did a great job last season... but you have to consistently show performances like that if you want to be world-class".
He continued: "People are so quick to say someone is a fantastic player and deserves to win the Ballon d'Or after just one good year. I disagree with that.
"For me, you have to prove yourself over 10, 12 or 15 years if you want to be among the best. Do that and then I will congratulate you.
"I might have been out for nine months or so due to injury, but I have almost always been playing at the highest level over the past 10 years or so."
Griezmann has since issued a measured response to the Bayern Munich winger, telling Eurosport: "I will continue working. As he says, we'll see at the end what kind of player I am, but I know I'm on the right track.
"Everyone has an opinion. To my father, for example, I'm the best player in the world. To the fans of Atletico, too."
Atletico president Enrique Cerezo also had his say on Thursday, commenting: "Ribery is a little older. We must forgive him because he was a great player."
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