Griezmann has been heavily linked with a summer move to Camp Nou following a number of public declarations from staff and players at the club suggesting that they would like to see him join the Spanish champions.
The 27-year-old recently insisted that he is still undecided on his future, but Messi has now joined the calls for the France international to swap Madrid for Barcelona.
"He's a great player and we would understand each other. He's going through an exceptional moment in his career," Messi told Mundo Deportivo.
"It's not yet 100%, but the more quality we have, the better. We want to be the best team in the world, and for that we have to look at the best."
Griezmann, who is understood to have a €100m (£87.9m) release clause in his contract, scored 29 goals across all competitions for Atletico last season.