Abate was heavily linked with a move to Fiorentina over the summer.
“I am here to prepare myself to the best of my abilities, and will be ready whenever the coach calls me up,” Abate told Milan Channel.
“I have changed roles compared to last season under Leonardo, and I think it is normal that I make some errors, but I hope to do well.
“I am working hard every single day. Under Leonardo, we played well, and were in the race to win the Scudetto when nobody predicted we had a chance.
“This season, we can do better, as many champions have arrived. We must not hide from the fact that we can finish near the top.
“Someone wrote that I was on the market, but I have always been calm. [Vice-president Adriano] Galliani and the coach [Massimiliano Allegri] want me, and I hope to stay for a long time.”