Redmond has signed a four-year deal at Carrow Road to be reunited with former Blues boss Chris Hughton.
It is believed the Canaries will pay an initial £2million for the 19-year-old, with a further £1.2million potentially due in add-ons.
Redmond was one of the bright spots of an otherwise disappointing European Championship campaign for the England U21 side this summer
He told his new club's official website: "I'm very happy. It looks like a fantastic club, I'm just looking forward to getting started now.
"I know that there is only one football club in this city and it has an excellent fan base. The people around the football club and in the city that I have spoken to so far have attracted me to this club so much, it's made me realise just how much this club means to the people of this city."