Reports according to the Express suggest that the Spanish giants are seeking to replace Iker Casillas - who left to join Porto, with another top class shot stopper.
Manchester United’s David De Gea has long been linked with a move to Madrid but it now appears Los Blancos would be switching attention to Ospina.
Everton have been in the fore front to sign the Colombian International from Arsenal following Petr Cech’s arrival from Chelsea but may now have to battle the Spanish money bags for his signature.
The Toffees were reported to have tabled a £5million bid to sign Ospina.
Real Madrid president Florentino Perez is reportedly said to be keen on the deal and talks could open as soon as possible to secure a move.
Turkish side Fenerbahce have also been linked with a move for the 26-year-old FA Cup winner.