Neesham suffered a blow during training on Thursday, which limited his involvement in New Zealand's recent three-day match against Mumbai.
Coach Mike Hesson said the injury was not serious, but he would not be considered for the Test starting on September 22.
"Jimmy was struck in the nets and has been in discomfort the last few days," Hesson said.
"It will take a few days for the bruising to go away, but it's very much a short-term injury.
"Jimmy won't be available for selection in Kanpur, but we expect him to be back to full fitness ahead of the second Test."
No one will be added to the 15 man squad.