According to the Mirror, the La Liga giants are lining up a £30 million offer for the 28-year-old, who signed a four-year contract extension last September.
However, the former Arsenal defender is reportedly unhappy with his life in England after his split from wife Cheryl Cole and would relish a fresh start in Spain.
Should the move happen, the defender would link up with his former boss Blues' boss Jose Mourinho.
Real Madrid are confident they can offer Cole a better deal than his current £120,000-a-week wage, and his strong relationship with Mourinho may be another incentive to move to the Bernabeu.
Cole is alleged to have sent a text message to his friends after England’s humiliating 4-1 defeat to Germany at World Cup 2010, which read: “I hate England.”
Steven Gerrard is also linked with a move to Real Madrid, although the appointment of Roy Hodgson may be enough to convince the Liverpool captain to stay at Anfield.
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