Chelsea have met the buy-out clause in the 25-year-old's contract with Atletico and he will now negotiate personal terms.
Costa was one of the revelations of the 2013/14 season, scoring 36 goals in 52 games for Atletico as they won the La Liga title for the first time since 1996.
He had to limp out of Atletico's Champions League final defeat to city rivals Real last month after just nine minutes but is now training with the Spain squad ahead of the World Cup in Brazil.
Blues boss Jose Mourinho made a striker his summer priority after Fernando Torres, Samuel Eto'o and Demba Ba all struggled for goals last season.
Costa was one of Atletico's scorers in the semi-finals of the Champions League when they came from behind to knock Mourinho's men out with a 3-1 win at Stamford Bridge.