The Portugal international, 32, has been linked with a return to Manchester United and Mourinho has stated previously he intends to make a big signing ahead of next season.
Reports in Spain claim Ronaldo will join United if Real are able to persuade Brazil forward Neymar to move from Paris Saint-Germain.
United sold Ronaldo to Real for a then-world record £80 million ($114 million, 92 million euros) in July 2009 and while Mourinho did not want to heighten speculation, he gave United fans a glimmer of hope on Thursday.
“I think at the moment Real Madrid is with him now. I think I should be the last one to add some fuel to the fire,” the United manager said.
“Madrid is on fire. The results are not good, and it’s a club where I worked for three years, I care about the club and I am the last one to add fire.
“So to put some water on the fire I think Cristiano is the kind of player that every manager wants, every club wants,” the Portuguese boss added.
“But only one manager can, only one club can have — (Zinedine) Zidane and Real Madrid, that’s my feeling.
“We want to improve all the time and get better and better. We want to achieve big things. We are not going to stop until we do that.”
Mourinho meanwhile believes Alexis Sanchez has left a “fantastic club for a giant club” as he prepares to hand him a United debut against Yeovil Town in the FA Cup on Friday.
The Chile forward, 29, joined United from Arsenal in a swap deal which saw Armenia playmaker Henrikh Mkhitaryan move in the opposite direction.
Sanchez will be United’s squad for Friday’s fourth-round tie at Huish Park and Mourinho could not resist a veiled dig at Arsene Wenger’s side as he considered the merits of the transfer.
“Manchester United made a fantastic deal,” Mourinho said.
“I think Arsenal also made a fantastic deal… I got a fantastic player, and Mr Wenger got a fantastic player.
“Alexis changed from a fantastic club to a giant club. Henrikh Mkhitaryan changed also to a fantastic club. So I think it was a great deal for everyone.”
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