The striker is reported to have a release clause of €100million in his contract, with Barca ready to bolster a forward line that already boasts Lionel Messi, Luis Suarez and Ousmane Dembele.
But Costa, who re-joined Atletico in the January transfer window, feels Griezmann can be persuaded to stay at the Wanda Metropolitano.
"He is taking a little time to say what is next for him and I think there is perhaps some doubt on his part," Costa told a news conference.
"What I'd say to him is that you have to be happy and you must feel that way because a happy player will keep getting better. But if you think you must move to be happy, then you will have to leave as players need that comfort and happiness.
"He knows that we love him very much and that he is the figurehead here, a very important player. If he stays with us, that would be brilliant.
"If he leaves, we also must be happy because he's given us a title this season but, in the end, I think he'll stay."
"Many things have happened since Brazil," Costa added. "The [2013-14] Champions League final, the injury... I did not play the World Cup in the best conditions.
"I am looking forward to this year, now we've got a different coach [Julen Lopetegui], other players... I'm enjoying it.
"Last time was not good, but now I'm very comfortable."