Thomas Tuchel revealed last weekend that the 29-year-old defender will leave Chelsea when his current contract expires this summer.
Real Madrid are now understood to have reached an agreement with Tony Rudiger’s representatives to move the defender to the Santiago Bernabeu as a free agent in July.
Although there would be no transfer fee involved in the agreement, Marca reports that a contract to hire a German defender player could cost them around $ 59 million (£ 50m).
The report added that Tony Rudiger’s salary and bonuses would be equivalent to the region of 11 million euros a year on a four-year contract, and 15 million euros would be spent on commissions.
Tony Rudiger, who returned to Chelsea on Thursday in a 1-1 draw with Manchester United, has played 197 games for the Blues in all competitions and has won five major trophies.