Financial details of the deal were not disclosed, but reports suggest Chelsea had paid an initial €20 million plus add-ons for the full-back.
The 28-year-old has put pen to paper on a five-year contract until 2031 and becomes the latest addition under new head coach Xabi Alonso.
The move fills the void left by Marc Cucurella, who joined Real Madrid earlier this summer in a €60m transfer.