Hazard, 28, revealed earlier this month he had made a decision on his future, but whether he is staying at Stamford Bridge or leaving remains unclear.
The Belgium international is contracted until 2020, yet has been linked consistently with a switch to Madrid.
Mourinho, his former coach and an ex-Madrid boss, is unsure what Hazard will do, but he believes the attacker is capable of starring at the Santiago Bernabeu.
"Eden's ambitions... playing at Chelsea his whole life or going out to a giant like Real Madrid?" he told DAZN on Sunday.
"I can't tell you, because I haven't talked to him for quite a while.
"Let's talk about talent. Does he have the talent to play for Real Madrid? He does. Does he have the personality to wear a super heavy shirt and play in front of a crowd like the Bernabeu? He has personality, yes."
The forward made his spot kick in Sunday's Carabao Cup final for Chelsea, but it was not enough as the Blues fell to Manchester City in penalty kicks.
Chelsea are back in action midweek when Hazard and the Blues take on Tottenham.
The Blues currently sit sixth in the league, three points behind fourth-place Arsenal while maintaining a game in hand.