Chelsea offered Ruud swap deal
The Daily Mail suggests the Spanish giants are intent on signing the Portuguese centre-back during the January transfer window and are willing to allow Van Nistelrooy to go in the opposite direction.
Los Galacticos are desperate to add reinforcements at the back following the knee ligament injury sustained by Pepe last weekend, which will keep the defender sidelined for up to six months.
The report also indicates Madrid president Florentino Perez has been a long-term admirer of Carvalho and attempted to sign the 31-year-old in 2004 prior to his £20 million move to Stamford Bridge.
Meanwhile, Van Nistelrooy looks destined to leave the Bernabeu after making just one league appearance this season.
The former Manchester United hitman has been heavily linked with a return to the Premier League and would provide cover for Didier Drogba and Salamon Kalou during January – who will be away on international duty at the African Cup of Nations.