The 28-year-old is to undergo a medical after it was announced on Thursday that the Spanish club have re-signed him for a fee believed to be in the region of £58m.
Costa moved from the Vicente Calderon to Stamford Bridge in 2014 and scored 59 goals in 120 games to help Chelsea to two Premier League titles.
However, the relationship between Costa and manager Antonio Conte has deteriorated this year, and the Spain international ended up in exile in Brazil rather than training with the team.
Costa told Marca: "It's not the way I wanted to leave, ever. I have a 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.
"I wanted to return [to Atletico]. I'm not going to be ungrateful to Chelsea, where I was also very happy, playing in a team that was great. Atletico is my home and I'm here.
"I'm very happy. It's taken a lot of time but it has a happy ending. Now, I'm waiting to see my teammates and train. I'm looking forward to giving my best as always, and hopefully things go well.
"It always makes you a little nervous when time is passing by, but [Atletico chief executive] Miguel Angel Gil has made a great effort and I'm grateful to him, like everybody at the club."
Costa will remain in limbo until January as Atletico are currently unable to register new players due to a FIFA transfer ban.