Mourinho was axed on Monday on the eve of Tottenham's Carabao Cup final against Manchester City this week.
Real president Florentino Perez remains a fan of Mourinho and maintains regular contact.
And with doubts over Zinedine Zidane fulfilling the final year of his Real contract, the Athletic reports Florentino has Mourinho on top of his shortlist of replacements.
Just Monday, Florentino said: "Zidane is the best coach we have ever had. He's a legend. He has a personality that is almost like mine, he can't stand press conferences. He doesn't say anything, he's happy… As he says, you never know. He has a contract in force."