After being told by Blues boss Antonio Conte that he was no longer needed at the London club, the combative forward remained AWOL despite no transfer coming through before the end of the window.
However, after returning to Madrid, the 28-year-old made it clear that he was pleased to have his protracted transfer saga come to an end by returning to the club he left for England in 2014.
"I'm not going to be ungrateful to Chelsea, where I was also very happy, playing in a team that was great," the BBC quotes Costa as saying.
"Atletico is my home."
Costa is expected to have a medical with Atletico in the coming days but will not be able to play for the side until the new year, with the Spanish club's transfer ban ends.
Despite the delay in his playing career, the man who scored 20 league goals for Conte's side last season made it clear that he would not be harbouring any hard feelings toward his former employers.
"It's not the way I wanted to leave, ever," Costa said.
"I have special affection for the people at Chelsea. I'm not a person who is going to smear the image. They know the person I am."