The Sri Lanka team arrived in Christchurch on Friday, a week before the first test begins, but 37-year-old Herath did not travel due to the injury which he suffered in a recent one-day game against England.
Sri Lanka captain Angelo Mathews said there was faint hope Herath would play in the match at Hagley Oval but it was unlikely.
Mathews said Sri Lanka would be "playing without Mahela for the first time in about 18 years and that's going to be a challenge for us. But there's pressure on all of us anyway. Playing against New Zealand in New Zealand is going to be tough."
AP