The 28-year-old has refused to return to Stamford Bridge after being frozen out of Blues boss Antonio Conte's first-team plans and failing to secure a transfer away over the summer.
Costa is thought to have relocated to Madrid in recent weeks in an attempt to force through a return to his former side, who will be able to officially sign him when their transfer ban is lifted in January.
According to AS, Diego Simeone's side have now reached an agreement with Chelsea over a deal worth €60m (£53m), which would make Costa the club's new record signing.
The publication claims that Costa's medical will now go ahead in the Spanish capital this afternoon, with an official announcement likely before the weekend.
While Costa is not able to play for Atletico until January, he is expected to join his new teammates in training for the next few months.