The 19-year-old joined City from Italian side Fiorentina on transfer deadline day.
"I'm so happy, I just can't put it into words," Nastasic told Mondo.
"I have become part of a great club. Manchester City are going places and I'm now part of it. It feels truly great.
"I was really surprised when City called me and then everything happened so quickly I couldn't believe it. I'm living my dreams."
He added: "I hope that I will not need a lot of time to adapt but I know that it is not the same to play in Italy as England.
"This is a big step ahead in my career. I'm only 19 years old."