The highly-rated 19-year-old midfielder has penned a three-and-a-half-year deal with the option for a further year with Canaries, subject to approval.
Maddison has immediately been loaned back to League One side Coventry for the remainder of the season as the Sky Blues continue to chase promotion to the Championship.
Norwich boss Alex Neil was delighted to have captured the talented youngster, telling the club's official website: "We're delighted to sign James as he is a fantastic prospect who has already made a mark at Coventry at such a young age.
"He has real potential to develop, making this deal one we're really pleased to get over the line.
"James will return to on loan Coventry for the rest of the season to continue his development before joining up with us on a full-time basis in the summer."
TRANSFER NEWS | #ncfc sign @Coventry_City midfielder James Maddison, subject to approval: https://t.co/DvlnCzx0fspic.twitter.com/HyAD2U8Tpt
— Norwich City FC (@NorwichCityFC) February 1, 2016