The 24-year-old left-armer was struck on his right wrist while training with the Northern Districts and is also doubtful for the second match of the two-Test series.
Santner, who has played 10 Tests for the Black Caps, showed promise during the recent tour of India where New Zealand were blanked 3-0 in the Test series and lost the following five-match ODI series 3-2.
Mohammad Amir looking forward to New Zealand, Australia tour
“After such a strong tour of India, it's really disappointing for Mitch to pick up this injury ahead of the first match of the home summer,” said New Zealand coach Mike Hesson in a statement.
“We're unsure whether Mitch will be available for the second Test at this stage, but we'll be monitoring his recovery closely and look forward to having him back,” he added.
Hafeez still out, Sharjeel in Pakistan squad for New Zealand tour
The first Test against Pakistan begins on November 17 in Christchurch, with the second Test in Hamilton from November 25.
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