The striker is currently away with the Brazilian national team ahead of the World Cup, but is believed to already have completed a medical.
Reports suggest that Chelsea have coughed up around £32 million - his release clause fee - for his services, and now Sky Sports claim that the deal should be announced on Tuesday ahead of the Brazil's World Cup opener against Croatia on Thursday.
The quote a source as saying: "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."
Costa scored an impressive 27 goals in La Liga to help Atletico take the title from Barcelona and Real Madrid last term.
That's just the type of goal scoring prowess that Chelsea were lacking last season. They missed out on the title by just four points, with boss Jose Mourinho regularly pointing to his lack of firepower as they reason they failed to keep up with the likes of Manchester City and Liverpool.
Costa is expected to be the first of many new arrivals at Stamford Bridge, with his team-mates Filipe Luis and Tiago set to follow him, while Cesc Fabregas is reportedly on the brink of a switch to west London.
Do YOU want to write for GiveMeSport? Get started today by signing-up and submitting an article HERE: https://www.givemesport.com/writeforgms