The England defender is out of contract at the end of the season and so numerous clubs are monitoring the situation, however Madrid are set to make their move when the window opens this week.
Cole has not been offered an extension to his current deal, along with long serving players John Terry and Frank Lampard, but Jose Mourinho does want to keep the core of his team together.
The 33 year old is set to be offered a one year deal at Stamford Bridge, however the former Arsenal man will want more assurances than that and so Madrid could open contract talks over a two year deal.
Cole can begin talks with Real on January 1 as they are a non-English club and he is out of contract in the summer, leaving Chelsea with a big decision to make in the coming days.
The usual ever present has been out of the team in recent weeks after an injury, although when fit Mourinho has still decided to play Cesar Azpilicueta at left back although a move for Southampton star Luke Shaw could be on the horizon.