The Daily Mail says Gauld, 17, was the subject of a scouting mission by a Real scout as the youngster emerged from the bench for Wednesday night's 4-1 League Cup win over Partick Thistle at Tannadice.
United midfielder John Rankin insisted Gauld is the Real deal.
"Ryan has a massive future," said Rankin. "The talent in the boy is incredible. If he is being linked with Real Madrid at that age, then it wouldn't surprise me - it really wouldn't.
"He is so level headed and if anyone deserves it, it would be him. I don't know if someone is hyping it up or someone is just trying to make a story.
"But I hope for the sake of the boy it's true.
"It certainly would not surprise me if he moved on to a big club. You want to see him doing well.
"He is so grounded and I hope for his sake and for the sake of the club - and a right few million - that it's true."
Real already have teenage Scots prospect Jack Harper on their books.
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