Atletico Madrid are awaiting a response from Chelsea after lodging a €60m (£54m) offer this week, say reports in Spain.
According to Marca, Los Rojiblancos have made an official offer for the 28-year-old to the English Champions and are hopeful of concluding the official arrangement soon.
The news follows on from reports that the Spain international did not board his scheduled flight from Sao Paulo to London earlier in the week.
He will not be able to play for Diego Simeone’s side until January as the transfer window will not officially open until then, but the clubs could confirm the deal in advance.
Despite being included in the club’s Premier League squad, it is not thought the striker will play for Antonio Conte’s team again after a text message earlier this summer instructing the player to find a new club.
Having left Atleti for Stamford Bridge in 2014 after winning La Liga, Costa has repeatedly expressed a desire to return to Madrid throughout the summer.