BBC Sport says the goalkeeper has re-signed for City from Burnley for a fee of £31m plus bonuses - a record for a British keeper.
Trafford, who has signed a five-year deal, told City's website: "It’s brilliant for me. It was one of my goals since I joined when I was 12 to play for Man City. I knew my journey would look different.
"Throughout it I always believed I would be coming back and when I left on loan when I was 18 or when I left for Burnley when I was 20, I always believed that one day I would be back and I worked to achieve this moment.
"I always believed that I’d play for City and I knew the only person that could do that is me. I always believed that I’d get to this moment and now I’ve got this moment, I’ve just got to keep going."
Amazing final season with Burnley
On helping Burnley win promotion last season, Trafford also said: "The season as a whole was amazing. The group that we had, the manager, the staff, it was brilliant. The records are nice but the main thing is we got promoted and everyone played their part in achieving that.
"Although in the Championship, do you think playing with the pressure to win every week has helped you coming back to City now where the expectations will probably be similar?
"It’s helped me, but throughout my time away from the Club I’ve had all different experiences. I’ve had the relegation with Burnley, I had the promotion.
"I had the one at Bolton where we were trying to get in the play-offs. I’ve had all the experiences and nothing will be a surprise to me."