Barrett suffered a calf injury during the first test in Dunedin last Saturday.
The lock was replaced midway through the second half of the 31-27 win to the home side.
A scan on Monday revealed a tear.
The All Blacks are yet to make an announcement about any possible replacement but the Blues' Patrick Tuipulotu would be the obvious choice, RNZ reports.
Barrett, Fabian Holland and Tupou Vaa'i were the second row options in the first test, with Vaa'i starting at No 6.
Assistant coach Scott Hansen said Barrett was a big loss.
"He's a massive part of the team. We're obviously disappointed to lose the skip, but we've got to go forward. We trust our depth, there is an opportunity for someone else to step up."
Hansen said they were yet to decide if they would call in a replacement.
"Razor (coach Scott Robertson) will go through that process and do what is best for the squad."
There was no time frame for Barrett's return.
"We'll just take it week for week, he'll be keen to get back as quickly as possible, but we obviously have to do what's right for him in the injury process."
Ardie Savea will take over the captaincy for this weekend's second test in Wellington.
