However, Diego Simeone's men face a significant obstacle in that they must overturn a transfer ban to allow the move to go ahead.
The situation is complicated by the Spanish side's transfer ban, but they are conducting their business as normal and are hopeful they can overturn the ruling at the Court of Arbitration for Sport, where a resolution is expected in the next week or so.
Muddying the case further is the complicated relationship between the two clubs - the Blues threatened to report the Spanish club to Fifa last summer over their aggressive pursuit to re-sign Costa.
Costa still has a close personal bond with Diego Simeone at his old club and he has not adapted well to life in the English capital. He opened up to reporters on Saturday after scoring in the defeat to Arsenal, as he claimed Chelsea might consider cashing in on him.
"I will only leave Chelsea for Atletico," Costa said. "If not, I will stay here. I'm not interested in other clubs. I have a contract and no intention of leaving, but if there are changes to be made that might reduce my chance.
"If the coach won't count on me anymore or is to give space for another striker, I know that I'll have to leave. Everyone know what my preferred club is, there's nothing to hide."
Atleti are very close to agreeing terms with Lyon striker Alexandre Lacazette, and Kevin Gameiro will make way if everything goes ahead as they plan.
For their part, Chelsea are unlikely to receive as much from Atleti as they were expecting to from Tianjin Quanjian, who no longer look well positioned to pull off a deal.
The CSL outfit were preparing to make a £76 million bid for Costa in the summer after failing to agree a deal with Chelsea in January, but the Chinese Football Association have changed their rules to make the transfer twice as costly.
Diego Costa scored 22 goals this season for Chelsea. His highest return for the club. #CFC— Nizaar Kinsella (@NizaarKinsella) May 27, 2017
Costa's head had turned after a huge contract offer but the authorities in the CSL have imposed a 100 per cent tax on transfers, which will come into effect when the next transfer window opens.
Chelsea had contract negotiations with the striker open at the time of the bid from China but they have since come to a standstill, following the player's falling out with coaching staff.
Chelsea want to make a major statement by signing a striker in the summer, with Romelu Lukaku and Alvaro Morata at the top of their shortlist, but the latest developments mean that the club may need to reevaluate its business.