A statement on Atletico's website confirmed the clubs will work over the next few days to complete the transfer.
Last year it was Manchester United who looked to be leading the chase for his services only for Griezmann to end up signing a one-year contract extension with Atletico, tying him to the club until 2022.
Just days before the start of the World Cup, Lionel Messi dropped a bombshell on the football world by saying his worldwide career may be over after the tournament.
Griezmann is expected to announce his future before his nation's World Cup campaign begins against Australia on Saturday.
However, Griezmann disappointed plenty of football fans by saying: "I am very sorry, but today I am not going to give my decision".
"I know there's a lot of expectation, I know many were waiting for it, but today is not the day (to speak about my future)".
But, according to "Canal + Francia", Griezmann has chose to stay with Atletico and lead the club's project.
Griezmann was close to closing a move to Barça after months of talks once his release clause dropped to 100 million euros on July 1.
So despite the temptation to play with Lionel Messi at Barcelona, most signs point to Griezmann staying put.
Also playing against a potential move to Barcelona, sources added, is that the Catalan club have told Griezmann they want him to play out wide - something is he is not so keen to do on a regularly basis.