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Kovacic made 51 appearances as the Blues secured a return to the Champions League by finishing third in the Premier League and winning the Europa League.
Chelsea are currently banned from registering new players for the next two transfer windows by FIFA, but were able to make the deal permanent as they already held Kovacic's registration from last season.
According to reports, Chelsea will pay Madrid around £40 million ($51 million) for the transfer.
"I really enjoyed my season on loan with Chelsea, I feel comfortable at the club and like London and the Premier League very much. I am very happy to be able to join permanently," said Kovacic in a club statement.
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Chelsea also confirmed on Monday they will not be taking up their option to buy on striker Gonzalo Higuain.
The Argentine scored just five goals after signing on loan in January and will now return to Juventus to again join forces with Maurizio Sarri, who left Chelsea to take charge of the Italian champions last month.