According to Ozgur Sancar of AS, the Turkish outfit are "confident" of snapping him up until the winter transfer window because Atletico cannot register players until then.
Costa can still join Los Rojiblancos this summer but would be required to sit on the sidelines until January if he did.
According to IUSport's Ivan Palazzo (h/t COPE's Isaac Fouto, via Yahoo Sport's Andrew Gaffney), he would need to be loaned to Besiktas from Chelsea rather than join Atleti and then temporarily head to Istanbul:
The Turkish club's supporters have done their part to convince Costa to join:
There are details that would need working out before the loan can be agreed upon, though, as the 28-year-old would not be able to feature in the UEFA Champions League for Atletico if he were to appear in the competition for Besiktas.
Further, per Sancar, they can't come close to covering his salary, which amounts to €8 million annually, even he'd only be there for less than half a year.
That would require Chelsea to supplement his wages until January, something they're likely keen to avoid because if they want him off their books as soon as possible. In addition, a loan deal could prevent them from receiving a fee for the striker until January.
The three clubs may need to come together to agree on a deal that suits all parties.
If they can, and Costa does head to Besiktas, he would significantly boost their prospects for the first half of the season. He netted 20 Premier League goals last season and could be even more prolific against the lower calibre of opposition in Turkey.