The 25-year-old Spain striker has long been linked with a move to Stamford Bridge, and the switch has now been confirmed by the Blues.
Chelsea will pay around 44 million euros for the striker, although a significant portion of the transfer fee includes fees paid to agents.
The Brazil-born Spain international will also triple his wages at Stamford Bridge, with Chelsea set to hand the 25-year-old a five-year contract worth 194,000 euros a week.
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The move has long been discussed, with Costa confirming the pending agreement to Goal in June: "Why would I have chosen to play for Chelsea? It is a thing that is about to happen. Things are going in a direction and, if they keep going as they are going, it is the team I will play for in the next season.
"I've wanted to play in the Premier League for a while. Chelsea are one of the greatest clubs of the world so there are many factors that make you take a decision like this."