The Spanish club's president Ramon Calderon revealed his desire to land Ronaldo after failing to do so once before when the 22-year-old snubbed him to sign a new five-year deal at Old Trafford.
United are certain to turn down any offer, but Real see Ronaldo as the player to return them to Champions League glory.
Calderon said: "Who wouldn't want to have in his team a player of Cristiano Ronaldo's amazing talent? He is one of the best in the world."
They have been encouraged by the Portugal star's recent admission that he wants to play in Spain while still at his peak.
A fortnight ago, at FIFA's World Player of the Year gala in Zurich, Ronaldo told Mirror Sport he longed to experience La Liga - despite signing a contract extension in March to become the highest-paid player in United's history on £120,000 a week.
"I've said on many occasions I'd like to play in Spain," he said. "It's a fantastic league, very attractive, where they play a beautiful game.
"It's my dream to play in Spain. Whether or not it will happen, I don't know. Sometimes I want something but it's not possible.
"Only God knows what will happen in the future. I'd like to find out the difference between playing in England and Spain."
United are adamant they will reject Real's advances. But with the Spaniards said to be ready to increase their bid if necessary, the Old Trafford board may have a tough decision to make come the summer.
Real's No.1 target had been AC Milan's Brazil star Kaka, but they believe they have a better chance of luring Ronaldo.