Manchester United are also interested in acquiring the Frenchman's services, but Barcelona now appear to be in pole position to land his signature.
Many expected Griezmann to move to Old Trafford this summer, be he opted to stay with Atletico out of loyalty and sign a new contract with the club.
The 26-year-old is yet to hit form this season and cut a frustrated figure Atletico's recent La Liga clash with Deportivo La Coruna.
He did, however, score for France against Wales on Friday, meaning he will be full of confidence heading into this weekend's Madrid derby.
Barcelona believe they have the upper hand on United boss Jose Mourinho, given that they can offer him a spot alongside Lionel Messi and France teammate Ousmane Dembele.