The website reports that Los Blancos have already asked Gareth Bale about his Welsh international teammate and that the La Liga club are seriously considering a move for the 27-year-old.
The Gunners withdrew a contract offer to Ramsey last month, meaning he will be able to sign a pre-contract agreement with a foreign club as early as January.
Real are looking for possible alternatives to Luka Modric and Toni Kroos, who are being tipped to leave the Bernabeu at the end of the season.
Ramsey’s situation, meanwhile, remains uncertain. The midfielder is still a first-team regular under manager Unai Emery, but it seems his future lies away from the Emirates, with Manchester City, Manchester United and AC Milan all possible alternative destinations.