Scuffet was Udinese's first-choice goalkeeper in the 2013-14 campaign and has been strongly linked with a move away this summer.
Atletico, Chelsea and AC Milan are reportedly keen to acquire the 18-year-old's services.
"Simone Scuffet will remain at Udine," Scuffet's agent Claudio Vagheggi told www.tuttomercatoweb.com.
"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."
Scuffet, dubbed 'the next Gianluigi Buffon', recently extended his contract with the Serie A outfit until June 2019.
Udinese are owned by the Pozzo family, who also own Watford FC in England and La Liga side Granada.