Liverpool made a bid of £21.8m for the striker last week, a sum that was believed to provoke a release clause in his contract.
"I'm well here, I'm happy here," AS quotes the 24-year-old as saying.
"As I've been in Argentina and Uruguay, I've not heard anything about this. I'm happy here, if the club want me to be here, I'm going to be here.
"I think that the coach [Diego Simeone] counts on all of us, not just me. He knows how important it is to maintain the squad, the fans know it as well."
Costa scored 20 goals in 44 matches for Atletico last season, helping them finish third in La Liga.