The deal is still subject to the 28-year old Sierra Leone international receiving immigration clearance.
The Canaries confirmed Kamara has been granted clearance to play by the relevant footballing authorities and the deal has been agreed between the two clubs and Major League Soccer.
Norwich boss Chris Hughton said: "I'm delighted to bring Kei to the football club, one of Major League Soccer's best players over the last few seasons.
"There is no doubt it is a big jump to make into the Premier League, but it is an exciting challenge both for him and us here at the club to prove his talent.
"Kei also adds some valuable competition up front alongside the current striking options we have here at the club."
The 28-year-old, who requires approval to work in Britain from the UK Border Agency, was born in Sierra Leone before emigrating to the United States when civil war gripped the country and settling in Los Angeles.
Kamara has scored 31 goals in 98 games for Kansas City.
Norwich are in the midst of a terrible run of form, having lost five and draw one of their last six Barclays Premier League games, while they were knocked out of the FA Cup by non-league Luton at the weekend.