The Spanish club are hoping to rush through a deal before Costa’s scheduled return to pre-season training with Chelsea next week.
The Spanish international hitman was told last month by manager Antonio Conte that he was no longer in his plans.
Atletico are currently banned from registering players after their sentence for breaching FIFA rules on the signing of minors was confirmed last month by the Court of Arbitration for Sport.
Atleti had agreed a deal to sign Alexandre Lacazette from Lyon this summer but that deal collapsed and the player has now moved to Arsenal.
The disappointment on missing out on Lacazette means Atletico are now likely to focus on buying back Costa.
Chelsea's top scorer last season wants a return to Atletico, though he would not be able to play until early next year.
It has been reported that Atletico may sign Costa and then loan him to a Chinese club. The current Chinese transfer window ends on July 14.
Chinese Super League side Tianijin Quanjian were said to have offered £76m for Costa in January, but Atletico may be able to secure his services for around £40m.
Atletico sold Costa to Chelsea three years ago for £32m.