Griezmann decided against a move away from the Liga side last year due to a FIFA-imposed ban on Atleti signing new players.
The 27-year-old is once more being linked with Barcelona and Manchester United, and has acknowledged that he wants his future to be resolved before the World Cup begins in June.
Costa rejoined Atletico from Chelsea last year, although the registration ban meant he had to wait until January to officially return to the Spanish capital, where he has established a formidable partnership with Griezmann in recent weeks.
"He knows he's an important player for Atletico. He's still with us and we hope he stays," Costa told a news conference.
"I'd love it if he stays here, but each person has to look for what's best. For me, the best thing would be for him to stay.
"I think he's going to stay. When I was in Brazil, he used to call me and tell me to come to Atletico. I don't want to stay here alone now!
"Griezmann knows we love him at Atletico. I hope he stays with us. He hasn't told me he is going to leave."
Speculation has emerged of a potential shock bid from Atletico for Juventus star Paulo Dybala, if Griezmann should leave.
And Costa would love to see Dybala at the Wanda Metropolitano.
"I don't know anything," he said when asked about the rumoured meeting. "We all know he's an amazing player and we want the best at Atletico."