Griezmann has been strongly linked with a move to Camp Nou for much of the campaign, to Atletico's chagrin, after a switch to Manchester United failed to materialise last year.
Such reports have intensified in recent days, but Costa has urged the Frenchman to think long and hard about leaving the Europa League finalists.
The Brazil-born Spain international found himself in a similar scenario when joining Chelsea in 2014, a move that initially worked well but turned sour due to a breakdown in his relationship with coach Antonio Conte.
Costa counts himself lucky Atletico kept the door open for a return, but he does not think that would be an option for Griezmann.
"I left and I was happy, I won things and I had a new experience," Costa told reporters.
"I decided it with my family and, even though I am here again, I will not tell him what to do.
"He is already able to make a decision. I had the good fortune to go back [to Atletico] and he knows that if he makes the same decision, there is no going back. So, he must think about it."
Costa pointed out that Griezmann remains a key figure at Atletico and is eager for him to remain, though he does not think the France international is affected by the speculation.
"He's big enough to know what to do," Costa said. "That he feels important is fundamental and he knows that we want him here.
11 - Antoine Griezmann's goal vs Levante ended a sequence of 18 passes by Atlético de Madrid, the longest sequence for Atlético de Madrid at home in La Liga with Simeone as manager. Elaboration pic.twitter.com/UUtv1M5gqU— OptaJose (@OptaJose) April 16, 2018
"Whether he leaves or not, I'm very happy with him. Now he's just thinking about winning a title.
"He wants to make history with us. I see him very focused. I see him very well, very focused, very sharp.
"If he has something [that sets him apart], it's his way of being inside the dressing room. He's normal and I don't see him differently from when I arrived, and the others also see him very plugged in and just wanting to win. I'm sure that it [the reports] does not affect him and motivates him a lot more."