Cane came off in the fourth minute after a head-knock during the 28-12 win at Waikato Stadium and will miss the match against the Stormers in Cape Town.
He will fly over to South Africa later this week for the second match of the tour against the Cheetahs in Bloemfontein, pending a fitness test, the team said on Sunday.
Prop Nepo Laulala has been scratched from the tour entirely after also suffering concussion during the Bulls match.
Uncapped Aidan Ross has been brought in as a replacement in the 27-man tour squad.
Inside centre Johnny Fa'auli has been included in the squad but bracketed with Alex Nankivell, having been cited for a dangerous tackle on Bulls centre Jan Serfontein.
(Reporting by Ian Ransom in Melbourne; Editing by Sudipto Ganguly)
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