Both clubs announced a "preliminary agreement" on a transfer, with Atletico saying the clubs will work together to complete the details.
Lemar is now away on worldwide duty with France and both clubs will be keen to finalise the transfer before the start of the World Cup, which gets underway on Thursday.
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 heavily linked with Barcelona over the course of the season just gone.
So, after months of speculation, it had been suggested that the 27-year-old would finally make his future known when it was confirmed he would be appearing at a France team press conference.
However, Griezmann disappointed plenty of football fans by saying: "I am very sorry, but today I am not going to give my decision".
With the World Cup starting on Thursday, the clock is ticking.
He made his France debut in 2016 and has scored three goals in 12 global games.
"I know that many of you were waiting for that".
He then went on to say that he would not be revealing what that decision was, as the time was inappropriate.
Reported to be the Spanish champions' first choice target of the summer window, many have anticipated that the Frenchman will move to Catalonia once his release clause drops to €100m in July.