26-year-old Hart has been added to the Spanish giants list of summer transfer targets, as they look to shake up their goalkeeping department.
Iker Casillas has fallen down the pecking order at the Bernabeu in recent seasons, and knows that his long term future is in serious doubt.
There is even talk that the Real Madrid and Spanish icon could be offered to City in part-exchange for Hart.
Current Real Madrid number one Diego Lopez is yet to convince manager Carlo Ancelotti that he is the long term solution in goal, and is also a reported target for AC Milan.
Hart shares the same representatives as Gareth Bale, who made the switch to Real in a world record breaking deal last summer.
The England goalkeeper has bounced back and is now showing close to his best form again, having been dropped by manager Manuel Pellegrini earlier in the season following a series of high profile errors.
However, Hart revealed over the weekend he has still not gotten over his treatment earlier in the season, with the comments giving Real belief they could tempt him to La Liga this summer.
“The past is the past and we don’t know what lies ahead in the future – so it’s better to live day by day.”