Madrid-based daily Marca understands that Los Colchoneros' last club-record bid for the Brazilian-born Spanish international is worth an initial €55m and includes an additional €10m in add-ons (£57.3m, $77.6m).
It is suggested that Chelsea are yet to respond to that latest approach for Costa, who remains on a self-imposed exile in his hometown of Lagarto as an ugly and protracted stand-off with the defending Premier League champions shows no signs of reaching an amicable conclusion.
One issue that seems to be providing a significant hurdle to any potential deal is that of loyalty payments. Marca further report that Chelsea do not want to pay the player 5% of the original transfer fee and that Atletico are also unwilling to do so.
The fiery Costa's 20 goals helped Antonio Conte to capture the title in his first season at Stamford Bridge in 2016-17, but, with his future subject of intense speculation since a training ground row and lucrative January overtures from China, the 28-year-old was quickly informed via a now infamous text message that he was not part of his manager's plans for the future.
After initially being given extra time off to sort out his future and finding himself omitted from a pre-season tour of the Far East, Costa has now spent almost four months in his native Brazil, defying repeated requests from Chelsea for him to report back to Cobham for training.
The former Real Valladolid forward has made it abundantly clear that he only wants to rejoin Atletico and it was even suggested that his eventual return was a key condition of popular manager Diego Simeone's decision to put pen to paper on a new three-year contract earlier this month.
An agreement was unable to be reached before the Spanish transfer window closed on 1 September, however. Turkish giants Fenerbahce, who went on to borrow oft-maligned Dutch frontman Vincent Janssen from Tottenham Hotspur, later failed with a last-gasp push to sign Costa on a short-term loan deal despite first-team boss Aykut Kocaman suggesting that the move was close. Ronald Koeman's Everton were also said to hold an interest in his services.
Club president Enrique Cerezo recently refused to deny reports that Atletico, unable to register any new players until January courtesy of a two-window transfer ban, were close to re-signing Costa, joking during an interview with Cadena Ser programme El Larguero that it's "always good to hear good news like this".
He added: "We can not register players but we can sign. We want Diego Costa. Another thing is that we can bring him because there are always a number of circumstances for a negotiation to go through. If that is solved, Costa will come. If it is not solved, he won't join us."
It was previously suggested that Costa, obviously extremely reluctant to head back to London, had travelled to Madrid and intended to establish himself in the Spanish capital until he could seal a move to Atletico in the New Year.