The 28-year-old is out of contract at the end of the season with the Gunners, and the club do not plan on offering him an extension to his current deal.
The Wales international will be free to sign a pre-contract agreement with a continental club on January 1st, and Real are able to offer superior wages to land him.
Real Madrid may have a slight advantage over other potential suitors in luring Ramsey to the Bernabeu as they have fellow Wales teammate and good friend Gareth Bale to persuade him to join the La Liga giants.
Ramsey is in his eleventh year at Arsenal having signed from Cardiff as a 17-year-old. He has made a total of 347 appearances and scored 60 times.