The Spanish club are looking for a world-class recruit to replace the disappointing Alvaro Arbeloa this summer and believe Cole's personal problems could tempt him to quit the Barclays Premier League.
World class: Cole is a key member of the Chelsea defence
Chelsea's high demands are a barrier to the deal - with the Bluesthought to be seeking at least £25million for the 29-year-old - but theBernabeu giants will persist in their pursuit if given encouragement bythe player's camp.
Troubled: Ashley Cole and wife Cheryl, the Girls Aloud singer, have split
They could even tempt him with a 50 per cent increase on his current £100,000 a week Stamford Bridge deal, which still has another three full seasons to run.
There are suggestions that Cole could seek a move abroad to escape headlines revealing alleged affairs, revelations that led to a separation with wife Cheryl.
Madrid are riding high in La Liga, but still see their defence as inferior to Barcelona's. Arbeloa has not impressed since a £3.5m summer move from Liverpool and head coach Manuel Pellegrini sees Brazil international Marcelo as more of a midfielder.
Madrid director general Jorge Valdano has been charged with drawing up a list of potential replacements and the Argentine sees Cole as being both attainable and good enough.
Manchester United's Patrice Evra has also been identified, but there is no indication that the France defender wants to quit Old Trafford.
Arsenal's Gael Clichy is also on the list of alternatives, along with Fiorentina's Juan Vargas, Aleksandar Kolarov (Lazio), Andre Santos (Fenerbahce), Luis Filipe (Deportivo) and David Luiz (Benfica).
Madrid bid £18m for Peru international Vargas in the winter window, offering the alternative of £13.6m and Royston Drenthe, but Fiorentina resisted a deal.
Madrid have struggled to fill the left back position since Brazil legend Roberto Carlos left in 2007.
Disappointing: Real's summer recruit Alvaro Arbeloa
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