Throughout the summer, Costa has been involved in a dispute with Chelsea after head coach Antonio Conte allegedly texted last season's top goalscorer to inform the attacker that he did not feature in his plans for the current campaign.
That has resulted in Costa remaining in his native Brazil until a transfer to Atletico Madrid can be agreed, although his former club are banned from signing players until January 2018.
It has been claimed that Costa has returned to London in recent days - despite the prospect of no action with Chelsea - but it appears that the 28-year-old may be on the move before the end of the Turkish transfer window.
According to Hurriyet, Costa will fly to Turkey in order to complete a temporary switch to the club and to maintain his match fitness as the Spain international looks to win a place in his national team's World Cup squad.
The Turkish transfer window closes on Friday, leaving Costa with 36 hours to complete a move.