Maddison passed a medical and agreed personal terms with the Foxes earlier this week, having decided against a switch to Southampton in order to be closer to his Coventry-based family home.
Leicester and Southampton had agreed identical packages worth in the region of £22m with Norwich to sign the England U21 midfielder.
But after Leicester announced the completion of a deal that could rise to £24m in add-ons, Maddison said: “I’m really happy to be here.
“I had a good chat with the manager about football and how he wants me to play, how he likes to play and I was sold. We talked about the type of player I am, how I’m going to fit into his style of play and everything sounded really promising. I haven’t looked back since that phone call.
“Leicester just seems like the right fit for me. I’ve got a good feel about the club and I can’t ignore that, in my head and my heart. I felt like this was the right place for me to be, so I’m delighted and can’t wait to get started.”
Foxes boss Claude Puel said: “James is a wonderful player who is one of the most exciting young talents in English football. He made a big contribution to Norwich City last season and I’m delighted that he has decided to make the step up to the Premier League with Leicester.”
Fulham, Everton and West Ham were also understood to have been interested in Maddison, while he had also been scouted by top-four Premier League clubs Tottenham, Liverpool and Manchester City.
Maddison scored 15 goals last season and, with their parachute payments due to end this year, Norwich have been willing to cash in on their most valuable player.
Norwich have been seeking between £20m and £25m for Maddison, who has represented England at U21 level.