Norris is expected to be sidelined for about six weeks, with a replacement prop to be announced in due course, New Zealand Rugby said on Monday.
He is the fourth player to be sent home from the tour, following injuries to hooker George Bell, centre Billy Proctor and winger Caleb Clarke.
All Blacks coach Dave Rennie is dealing with injuries on multiple fronts before meeting the world champion Springboks at Ellis Park on Saturday.
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First-choice fullback Damian McKenzie has been struggling with an ankle problem that kept him out of the 50-19 win over the Bulls in Pretoria on Saturday.
Prop Xavier Numia has a stiff Achilles and scrumhalf Cortez Ratima took a blow to the knee.
Halfback Noah Hotham has been called into the squad as injury cover for Ratima, New Zealand Rugby said.