The Spain striker has already passed a medical and Chelsea have met the £32m buyout clause in his contract.
Costa has also directed Atletico to accept the offer and Sky sources understand that Chelsea are determined to get the deal done before the start of the World Cup, so that he can concentrate on the tournament.
"Chelsea have made an offer to Atletico Madrid in compliance with the buyout terms in Diego Costa's contract, and have been advised that the player has provided notice to Atletico Madrid invoking the buyout clause and directing the club to accept Chelsea's offer," said a club source.
"Chelsea look forward to entering into formal transfer documentation with Atletico Madrid as soon as possible so that the player can concentrate on the World Cup."
The Brazil-born forward scored 36 goals in 52 games last season as he helped Atletico win the La Liga title for the first time since 1996 and reach the final of the Champions League.
Costa limped off after just nine minutes of the final in Lisbon which was won by city rivals Real but the 25-year old is fit again and training with Spain ahead of the tournament in Brazil.