The Ukraine fullback moved to Forest last week from Arsenal, but only joined up with his new teammates on Thursday due to international week.
Zinchenko told the club's website: “I feel unbelievable. I have to say a big thank you for the welcome from everyone. There are so many good people around the club and I am so happy.
“It’s an amazing opportunity. I want to help the team achieve its goals and show myself that I can do good things on the pitch.
“I remember the day when this opportunity came about; I was so happy and excited, to be honest.”
On where he expects to play for Forest, Zinchenko continued: “I used to be asked what position is my best, but to be honest I’m ready to help my team wherever I can and wherever I’m needed, and I will always try to give my best.
“I want to make an impact on the pitch but also off it. It’s important to be part of the team and help the young players as someone with that experience already. Then we can push each other to achieve big things.”