The Spanish giants are looking to replace Santiago Solari, who they only appointed as their head coach last month, when the season comes to a conclusion.
And Diego Torres, who wrote a book on Mourinho’s first spell at the Santiago Bernabeu, has reported in newspaper El Pais that Real and president Florentino Perez are keen to bring him back.
They would look to pay him €20million (£18m) annually after increasing an initial offer of €15m (£13.5m). Discussions have taken place since October.
Real are also planning to appoint him on July 1, as that would mean avoiding a €5m (£4.5m) penalty fee to Manchester United.
Mourinho was dismissed as United manager on Tuesday morning after a chastening defeat at Liverpool on the weekend. He left with the club down in sixth, and has since been succeeded by Ole Gunnar Solksjaer on an interim basis.
But Real believe the Portuguese manager is the man to replace Solari.
The former Real B manager has suffered two 3-0 defeats at the hands of Eibar and CSKA Moscow after signing a deal to keep him in the post permanently until 2021.
A survey of Madrid members suggested that re-appointing Mourinho, who managed Real between 2010 and 2013, would be a popular move.
There is a belief that he has the ideal personality and ability to lead a club like Real.
Even Marcelo, who fell out with Mourinho during his spell at Real, has backed him as a manager. ‘It is a real pity because he (Mourinho) is a great coach,’ he said.
‘It is a real shame that he is without a club and it is not my place to say that he should return to Madrid, but I have a lot of appreciation for what he did for me at the club.’
He will be handed a significant transfer budget and influence over transfers.