The Brazil-born Spain international joined Galo in August on a short-term contract, having been granted his release from Atletico Madrid eight months earlier.
Costa's second spell in Madrid was far less successful on an individual level than his first, which had earned him a £32million move to Chelsea in 2014, with the striker generally impressing in the Premier League as he scored 52 goals in 89 matches.
His August switch to Atletico Mineiro provided Costa with the opportunity to regain fitness after several months away from competitive action.
He featured 19 times for the Belo Horizonte club and scored five times across all competitions, helping them to a Brasileiro and Copa do Brasil double.
Arsenal are said to be interested in signing the 33-year-old on a free transfer, with Mikel Arteta apparently keen for striker reinforcements after Pierre-Emerick Aubameyang was ostracised from the first team.
Costa offered no clues as to where he will end up, however, as he bade farewell to Galo.
"Today, officially, I say goodbye to Atletico," he wrote on Instagram.
"I would like to thank all [Atletico Mineiro fans] for all the support and affection I received from the first to the last day.
"Thank you so much for being part of my childhood dream of playing in Brazilian football and winning titles.
"I would also like to express my gratitude to the entire technical staff, medical department, employees and especially to my teammates who helped me daily, always doing everything to make me feel comfortable.
"The feeling that remains is gratitude to all of you Atleticanos. Thank you and I wish you all the success in the world!"
While Costa could potentially join Arsenal, the Gunners are expected to announce that centre-back Pablo Mari has left on loan for Udinese after struggling for first-team football following the arrival of Ben White.