The controversial striker has so far refused to return to Stamford Bridge after he was allegedly informed by manager Antonio Conte via text message that he was no longer wanted.
Costa has made it clear that if he is to leave Chelsea, he only wants to return to Atletico, where he spent four years of his career before moving to the Premier League in 2014.
Atletico are permitted to buy players in this window, but they are not allowed to register them until January 2018 due to a transfer embargo upheld by the Court of Arbitration for Sport.
According to The Mirror, the saga could be edging towards a conclusion as both parties are coming close to agreeing a £30m deal.
Costa has been linked with a short-term loan move to Everton, who are looking for a striker before the transfer window closes on Thursday.