Costa, who has played a starring role in the Spanish club’s run towards a possible La Liga/Champions League double, has long been linked with a move to London, with Jose Mourinho desperate to sign a prolific striker.
Mourinho suggested at the end of Chelsea’s final Premier League game of the season that he was keen to make his biggest splash into the summer transfer market as soon as possible.
The deal with Costa is not expected to be formally announced until early June, after the big two games he is fighting to be fit for.
Costa suffered a hamstring injury and missed Atletico’s 1-1 draw with Malaga on Sunday, a game in which a win would have sealed the Spanish title.
Atletico travel to Barcelona on Saturday for a match that will now decide where the trophy goes and seven days later they take on city rivals Real in the Champions League final in Lisbon.
Chelsea finished the 2013/14 season in third place, four points behind Manchester City and two behind Liverpool, with their goalscoring record seen by many as the reason for them failing to win a trophy.
The Blues scored 71 goals, compared to 102 for City and Liverpool’s 101, and their only player in double figures was attacking midfielder Eden Hazard, who managed 14.
Samuel Eto’o, Fernando Torres and Demba Ba scored only 19 goals between them, often leading to speculation Mourinho would make a big-money move for a player like Costa, who has scored 27 times in 33 appearances in La Liga this season.