The 18-year-old has established himself as the first-choice shot-stopper at the Stadio Friuli and has been widely linked with a move elsewhere.
Chelsea, Milan and Atletico have been among those credited with an interest in the teenager.
But Scuffet's representative Claudio Vagheggi insists that his client will stay in Italy and rejected an approach from the Spanish champions to do so.
"Simone Scuffet will remain at Udine," Vagheggi told Tutto Mercato Web.
"Atletico Madrid's offer was interesting and advantageous but we agreed with the club and with the player's family that the best thing for him is to have a progressive and gradual growth.
"The best place for him to continue to develop is at Udinese, a professional atmosphere which at the same time is familiar to him.
"After one year playing at the highest level, his growth must be gradual.
"We are convinced this is the best choice for him."
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