The 34-year-old left the Rossoneri earlier this northern summer to join Swiss side Sion after making more than 300 league appearances for the Italian giant.
Long-serving centre half Nesta followed Gattuso out of the Milan exit door, joining MLS outfit Montreal Impact after 12 years at the club.
Gattuso says those at the top wanted him to stay, but claims Allegri was keen to move the former Italy international upstairs.
He told La Gazzetta dello Sport: "If Alessandro Nesta and I made this choice, then it's because we felt that
"When someone like the coach makes you understand that he rather prefers you as a director, or that he wants you to be someone who helps out the others in the changing room, you know what then? I will leave. I'll shake your hand without feeling any resentment and without holding a grudge.
"I have an exceptional relationship with
"No, I don't regret leaving Milan. I had many other offers, also from Italian clubs, but I refused them because I really didn't feel like joining them."
Roma has vowed to ban any supporters identifiable as responsible for the racial abuse of AC Milan striker Mario Balotelli.