Griezmann has been strongly linked with a move to Barcelona in recent months, with the France international widely tipped to complete a transfer to the Catalan giants following the 2018 World Cup.
Costa, however, expects his Atletico teammate to stay at the Wanda Metropolitano entering the 2018-19 campaign.
"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 reporters.
"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."
Griezmann, who is believed to have a £88m release clause in his current contract, scored 29 times in 49 appearances for Atletico during the 2017-18 campaign.