It is understood that representatives of Monaco and Atletico met on Tuesday evening to discuss the transfer, with Lemar now involved with the French national team at the World Cup in Russian Federation, as reported by French news outlet L'Equipe.
France travel to the World Cup on the back of a surprising 1-1 draw with the US in Lyon on Saturday night, but Griezmann reckons there's no need for his teammates to panic, and says their preparation for Australia starts now.
Barcelona are reportedly ready to trigger Griezmann's €100million release clause.
Some heartwarming footage has emerged of a young Antoine Griezmann meeting his France heroes back in 1998.
"I'm sorry for the wait, but today is not the day".
Monaco have already sold Fabinho to Liverpool this month, and they are set to make further profit with the sale of Lemar.
And, speaking to TF1's Telefoot, Griezmann said the announcement will be made this week because France's opener is against Australia on June 16.
Griezmann, 27, continues to be linked with a move to LaLiga champions Barcelona after another fine season at Atletico.
"However, today is not the time, nor the place, to say what it is".
Atletico Madrid reached a deal in principle to sign France global Thomas Lemar from Monaco on Tuesday.
The winger will hope to feature for France at this summer's World Cup and build on his 12 caps and three goals for Les Bleus.
Dermot Corrigan is a Madrid-based football writer who covers La Liga and the Spain national team for ESPN FC.