Griezmann worked actively to bring Lemar to Atletico and is excited to play with him, sources said before the agreement was announced.
A statement on Atletico's website confirmed the clubs will work over the next few days to complete the transfer.
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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.
On Tuesday Griezmann revealed he has made a decision regarding a potential summer transfer to Barcelona, but isn't ready to make that decision public.
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With ongoing speculation linking him with Barcelona, Griezmann had said he would confirm his position before France's World Cup opener against Australia on Saturday.
Griezmann has been dedicating a lot of time to his decision, and reports claim that he's turned to French teammates while preparing for the World Cup for advice.
Monaco have already sold Fabinho to Liverpool this month, and they are set to make further profit with the sale of Lemar.
Both Barcelona and Manchester United have expressed interest in the forward, who has a release clause of €200m.
Since joining from Real Sociedad in 2014, Griezmann has scored 112 goals in all competitions for Atletico and finished each season as the club's top scorer.