The hulking blindside flanker, who has played 77 times for the All Blacks and won two World Cups, suffered the injury in Super Rugby action for the Auckland Blues two weeks ago and will have surgery on Friday.
Blues assistant coach Steve Jackson said the operation would sideline the 34-year-old for four to six weeks but was confident he would have time to prove his fitness before the first test against the Lions on June 24.
"He was feeling it and it was locking up on him," Jackson told reporters in Auckland.
"Potentially he could have played but it was causing discomfort and if it locked up during game time we would have had to replace him anyway.
"He'll get his feet up and normally you come back quick from those sorts of operations. I'm pretty sure he'll be fine for the Lions tour."
As in most positions, the All Blacks do not lack for cover at blindside flanker with Liam Squire, Elliot Dixon, Steven Luatua and the experienced Liam Messam all proven performers at test level.
Luatua, who will be moving to England at the end of the season, will switch from number eight to replace Kaino for the Blues against Australian conference leaders ACT Brumbies on Sunday.
Blues lock Patrick Tuipulotu, who started three tests for the All Blacks last year, will also miss Sunday's match in Canberra because of injury, having hurt his back in training.
"He's got a flare-up of a disk in his back," Jackson said.
"It's a time thing for him. It could come right tomorrow or in two or three weeks. We're not sure."
(Reporting by Nick Mulvenney, editing by Peter Rutherford)
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