The former Blues boss is anticipating to compete a move to the Bernabeu on Monday when he meets with Real president Florentino Perez.
The ‘Special One’ hopes to bring Cole and Inter right-back Maicon with him to Madrid next season, according to the Daily Mail.
The England international, who Mourinho brought to Chelsea from Arsenal four years ago, is valued at about £25 million.
Cole returned to action late in the campaign after breaking his ankle in February and helped Chelsea secure a Double this season.
The 29-year-old has four years remaining on his £100,000-a-week contract at Chelsea and the Blues will fight to keep him at Stamford Bridge.
Become a fan of Goal.com UK's Facebook fan page for all the latest news and insight into everything related to the beautiful game!