"About this issue, we want to thank him for what he did with this club. We wish him all the best for the future," he said.
When quizzed about his relationship with the player, however, the Chelsea manager was coy.
"Now it is not important, I am not interested in continuing to talk about this issue," he added.
"I am working with my players and am happy to work them them. This issue is not important."
Costa is set to return to Atletico, though is unable to take to the field until January.
But Conte did praise him for all he achieved during his time in the Premier League.
"We won together last season and for sure we want to thank him for his effort last season, and in the period he played for Chelsea," he added.