Fenerbahce claimed that they were "close" to signing Costa on a loan deal until January but it is unlikely to go through, with Turkey's transfer window closing on Friday.
The Istanbul club's manager, Aykut Kocaman, has insisted talks have been taking place but that a move isn't expected to happen.
"We were really close to sign Diego Costa but now we seem far away," Kocaman said. "The talks between the two clubs were going really well. But here we are."
Atletico Madrid is the most likely destination for Costa still, but the transfer window is closed and the La Liga side failed to come to an agreement with the Premier League champions despite continued talks.
Chelsea have registered Costa in their Premier League squad but not their Champions League roster as they aim to retain his value ahead of a winter sale to Atletico.
Costa is unlikely to play for Chelsea, with the relationship still strained between him and Antonio Conte after the 28-year-old became enraged by a text message that said that he was no longer needed at the club.
Goal understands that Costa already had agreed to leave Chelsea in January, but he was unhappy with the timing of the text message that came while he was with Spain on international duty.
Costa has never properly settled to life in London despite success on the pitch. He now has his heart set on a return to Madrid to play under Diego Simeone, who he counts as a close friend.
The Blues striker has done several interviews in Brazil revealing his feelings towards Chelsea over his situation. In one interview, he said he feels like he is being treated like a criminal.
The situation gets messier still as Costa has legal proceedings open against Chelsea for "discrimination", as explained by his lawyer in a statement to the news agency EFE.
Costa tried to leave Chelsea in January after a £76 million bid arrived from the Chinese Super League club Tianjin Quanjian, who were also offering to almost triple his wages.
Tianjin were going to sign him in the summer transfer window for the same fee but the authorities changed the transfer regulations to impose a 100 per cent tax on foreign transfers, effectively doubling the fee that it would take to bring Costa to China.
A resolution still doesn't look close between Costa and Chelsea, but he remains a Blues player.