The winger still did enough to secure a place in Didier Deschamps' France squad for the Federation Internationale de Football Association 2018 World Cup in Russia.
According to French sports daily L'Equipe, Atletico Madrid sporting director Andrea Berta visited Monaco earlier this month and has managed to strike a deal with the Principality club.
But Griezmann has remained tight-lipped on his decision, refusing to reveal his future destination just yet.
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However, at a national team press conference in Russian Federation, the 27-year-old repeatedly laughed off questions regarding a potential move before claiming he has made up his mind where he will be playing next season.
The duo are being interviewed at the same time, reeling off names of their favourite players as the presenter says: 'They are so cute we can not refuse them anything'.
He has scored three goals in 12 matches with the national team since his November 2016 debut.
One of the standout performers for Monaco during their title-winning season and UEFA Champions League semi-final run in the 2016/17 campaign, the technically gifted midfielder will now negotiate personal terms before completing a transfer to the LaLiga outfit.