Santner experienced pain in his right foot at the end of the second one-day international against Australia in Wellington and a subsequent scan had shown bruising around the bone.
"It's unfortunate for Mitch as he's been a key player for us so far this summer," Hesson said of the 24-year-old, who has impressed in all formats with his composed batting and left-arm spin.
"The positive news is that we picked it up early and hopefully Mitch can make a quick recovery."
Santner would continue to be monitored and a decision on his availability for the second and final test in Christchurch would be made at a later date, the team said.
The first test starts at Wellington's Basin Reserve on Friday.
Australia defeated New Zealand 2-0 in the three-test home series in November.
