Chelsea are desperate to offload Costa before Thursday's transfer deadline, with the striker currently AWOL in Brazil and refusing to return to training after being told that he is surplus to requirements at Stamford Bridge.
The 28-year-old has repeatedly expressed his desire to return to Atletico, where he spent four successful years earlier in his career, although he has also questioned whether the Spanish outfit are making a big enough effort to sign him.
Atletico were reportedly refusing to make an offer worth more than £26m until recently - some way short of Chelsea's rumoured £50m asking price.
However, the Evening Standard claims that Atletico could now be willing to pay as much as £41m for Costa, who scored 20 goals to help fire Chelsea to the Premier League title last term.
The move is complicated by an ongoing FIFA transfer ban which prevents Atletico from signing any new players until January, although Costa could spend the first half of the campaign on loan elsewhere.