Griezmann was heavily linked with a switch to Manchester United during the summer and was thought to be on the verge of joining the Red Devils until Atletico were handed a transfer ban.
That embargo expires in January, when Atletico will welcome Diego Costa into their squad amongst others, although any deal for Griezmann is expected to wait until the summer.
French website Le10sport claims that Barcelona are now frontrunners in the race for his signature, though, having held secret talks with the France international in recent weeks.
Discussions are now understood to be at a very advanced stage, with both parties close to agreeing a deal - although Barca must still negotiate an agreement with Atletico too.
Griezmann is understood to favour a move to Barcelona over United as it will enable him to stay in Spain, where he has played since the age of 14.