The 19-year-old came through the Birmingham Academy and played under Canaries manager Chris Hughton when he was manager at St Andrew's, which included a run in the Europa League for the Championship club.
"Last season was a good one for me in terms of goals and assists and just general performances. Hopefully I can continue the form that I left off with last season at Birmingham," Redmond said on the official Norwich website, www.canaries.co.uk.
"Norwich looks like a fantastic club, I am just looking forward to getting started now."
Birmingham are understood to have also agreed a fee with Swansea for Redmond, in the region of £2million with add-ons. However, the winger opted instead to move to East Anglia and reunite with his former manager.
Redmond added: "Chris Hughton gave me the chance to play and the chance to express myself every time that I stepped on to the field, and I feel like on the bigger stage he will do exactly the same thing."
Redmond, though, admitted on a personal level it was not easy to be leaving his home-town club.
In a statement on his Facebook page, the 19-year-old wrote: "It has been a very emotional 11 years for me at Birmingham City FC and I have loved every single minute of it. I really, really am going to miss it from the bottom of my heart.
"My nan who passed away last year always used to tell me: 'Never forget where you come from.' Well I came from Birmingham City FC!! ONCE A BLUE ALWAYS A BLUE...
"This club will remain etched on my heart and soul for life, I am just sorry that I had to leave, but I am looking forward to the next step on my journey as a professional footballer. Thank you so much to you fans for supporting me on my journey from schoolboy to young professional."
- Sports & Recreation
- Chris Hughton
- Birmingham Academy