Lemar is now on worldwide duty with France at the World Cup, and will link up with his new team-mates after the tournament once the deal is officially confirmed.
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.
While Manchester United have been linked with Griezmann previously, Deschamps said the forward was staying in Spain.
Lemar and Griezmann are a part of the France team that will begin their Group C campaign against Australia on Saturday, before facing Peru and Denmark in the group stage.
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Lemar has scored 22 goals and registered 32 assists in 127 games over the past three seasons for Les Monégasques since joining the club from Caen.
It is widely believed that the arrival of Lemar, big news enough in its own right, will also be the deciding factor in keeping Antoine Griezmann at the club.
Despite Morata's best efforts, he and the rest of the world will have to wait a little bit longer to discover Griezmann's decision over his future.
Barcelona had hoped to sign the 27-year-old on July 1, when his release clause drops to €100 million, but Atletico have offered him an improved contract worth a reported €23m a year.
He's now preparing for the World Cup with France but given how far along the Liverpool transfer was, there's every chance his head is still on Merseyside.