The Liga giants are eager to move the 31-year-old striker on after seeing him struggle to find a spark during his second spell in the Spanish capital.
He was a talismanic presence during his initial stint at Atletico, helping them to a domestic title triumph and Champions League final.
At that stage Costa was considered one of the most fearsome strikers in Europe, with Chelsea signing him during the summer of 2014.
Three productive years were spent in England before re-joining Atleti, but just 17 goals have been scored in a further 83 appearances for Madrid.
Diego Simeone has given the Brazil-born star plenty of opportunities, but the desired impact has come. With that in mind, and an expiring contract to take into consideration, potential suitors are being sounded out.
Atletico are eager to avoid losing Costa on a free in 2021 and would prefer to find a buyer in the current window - but his reputation and price tag have taken a considerable hit.
His agent is working with Atletico to find a taker, but no formal interest has been found yet. That is despite teams in the Premier League, Serie A and Ligue 1 being approached, with possible routes to Turkey and Brazil also explored.
That could further complicate what is already proving to be a drawn-out saga.
The financial implications of Covid-19 mean that clubs around the world have less money to work with in the latest transfer market.
Asking prices are being reduced as a result, while moving unwanted assets on can be tricky. Atletico are finding that out the hard way, but Costa will leave Wanda Metropolitano if a suitable destination can be found. Simeone could look to keep the powerful forward if forced into a corner, but the exit door is open.