McDermott was relieved of his duties on Friday evening to end widespread speculation about the former Reading coach's future, but the Italian owner has rejected theories that technical consultant Carbone will take over.
Speaking to freelance journalist Simon Austin, who enjoys close links with the club, Cellino said: "I have always said that Carbone will be in charge of the Under-21s next season and that doesn't change now.
"We did not want to go on pushing, pushing each other, and we found a gentleman's way to part. Brian leaves the club on good terms, he is a gentleman."
McDermott took over at Elland Road in April 2013.