Griezmann, who is away with the France squad preparing for the World Cup, has decided not to force through a move to the Nou Camp, according to ABC.
Barca have long been linked with Griezmann, who has a release clause of €100million in his Atletico contract.
The reports in Spain continue to say that Griezmann will see his loyalty rewarded with a new contract which would make him one of the highest-paid players in Europe.
His relationship with coach Diego Simeone is said to be a major factor in his decision to remain as well as the recent Europa League win in Lyon.
He bagged a brace in the 3-0 final victory over Lyon to take his season tally to 30 goals and 13 assists, highlighting his importance to the side.
Back in the Champions League next season, the Frenchman has bought into Simeone’s vision that they can once again challenge both domestically and in Europe.
With Griezmann hoping to play a starring role for France in Russia this summer, the forward wants to clarify his future before the tournament gets under way on June 14 and so both Barcelona and Atletico should have confirmation soon enough.
Didier Deschamps side begin their campaign against Australia on June 16 before matches against Peru and Denmark.