The Spanish champions have attempted to sign Ronaldo before but were snubbed by the winger, who instead signed a new five-year deal with United to become the highest paid player in the club's history on £120,000 a week.
However, Real president Ramon Calderon is more optimistic this time around that a deal can be struck with United's owners, the Glazer family, who would find it hard to resist a world-record fee, particularly as they are currently finding it difficult to refinance loans due to the global credit crunch.