Dan Gibbs of the Daily Express cited a report from Radio Estadio, which said the deal will comprises of a £41 million payment and £9 million in add-ons.
Costa has refused Chelsea's orders to return to train with the club's reserves, and the forward, who has spent the last few weeks in his native Brazil, released a statement explaining his decision, per Gibbs:
"We're still negotiating, they have to decide what's going to happen to me. I have two years of contract with them and I hope it is resolved as soon as possible because I need to play.
"The coach (Antonio Conte) does not want me for some circumstances.
"While they do not define my situation, I started the training on my own. I just will not stay in a place where they do not want me."
Costa has made no secret of his desire to move back to Madrid to rejoin Atletico, with whom he won the 2013-14 La Liga crown and the club's first league title in 18 years.
It was reported by BBC Sport back in June that manager Antonio Conte had informed Costa via text that he no longer featured in his plans moving forward.
Conte appeared in front of the media on Friday and was asked for his take on Costa saying he was being "treated like a criminal" by the Blues, per the Guardian's Jacob Steinberg.
The Italian's reaction to such a suggestion appeared to rubber-stamp the notion that Costa's time at Stamford Bridge is over, with Conte calling him "the past" and saying he wouldn't pick the forward if he were to return to west London:
B/R Football @brfootball
Antonio Conte dealing with the latest Diego Costa news in the best way possible 😂 https://t.co/2ZV0j7PTZj2017-8-18 14:06:35
If news of an agreement between Chelsea and Atletico proves to be true, he still won't be able to play for Los Rojiblancos until January 18 as a result of their transfer suspension, per BBC Sport.
Despite being told he can leave Chelsea, Costa hasn't shown a professional approach in his effort to engineer an exit, according to ex-Blues star Michael Ballack in a recent appearance on BBC Radio 5 live Sport:
BBC 5 live Sport @5liveSport
"You have to stay professional" Former #CFC midfielder Michael @Ballack isn't impressed by Diego Costa's behaviour. Your thoughts? https://t.co/3XH1s845rl2017-8-19 09:00:08
According to Tony Banks of the Express, Costa could have to stump up £3 million in fines if he fails to show up for training until January, having already been booked several weeks' wages from his £150,000-per-week contract.
Negotiations over personal terms with Atletico would likely fly by given Costa's desire to return, and his acquisition will reunite manager Diego Simeone with one of Los Rojiblancos' most devastating servants.