Cole has lost his first team place at Stamford Bridge this season, with Jose Mourinho opting to use Cesar Azpilicueta at left back since November.
Cole is out of contract in the summer, and there have been no indications he is set to sign a new deal.
The Spanish giants are ready to offer the England international a deal worth a reported £75k a week, with sources in Spain suggesting Carlo Ancelotti is keen to make the deal happen.
Ancelotti worked closely with Cole during his spell as Chelsea manager between 2009-2011, with the pair forming a close bond and enjoying the success of a league and cup double during his first season in charge.
Premier League leaders Liverpool are reportedly preparing an offer to keep Cole in the Premier League, while there are also rumoured to be a number of MLS clubs interested in the 33-year-old’s services.