The 33-year-old left-back has slipped away from the Blues’ starting XI in recent months after falling out of favour with Jose Mourinho on the back of some shaky displays.
This has alerted Real coach Carlo Ancelotti, who is said to be keen to offer the experienced England international an escape route.
The Italian managed Cole during his time at Stamford Bridge, and is known to be a big fan of the defender’s abilities.
Ancelotti sees the ex-Arsenal man as an ideal solution to his current problems on the left side of his backline, with Fabio Coentrao rumoured to be unsettled at the Bernabeu.
Cole’s current contract expires at the end of the season, which could see him available for a minimal fee next month.
Ancelotti is ready to offer the veteran regular playing time In the run up to the World Cup, which may appeal to him with Leighton Baines challenging for a starting role under Roy Hodgson.
Chelsea may move for Southampton’s Luke Shaw in January, which would further limit Cole’s playing time.