The Germany international rejected the chance to stay at Stamford Bridge beyond this season.
He will officially join Real on July 1 on a four-year deal understood to be worth €400,000 (£342,000) per week, per Sky Sports.
Blues boss Thomas Tuchel confirmed last month that Rudiger would not be re-signing with the club.
But the 29-year-old was flattered by Real's interest and will now move to the Spanish capital.
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