De Kock, whose belligerent knock of 91 in a 160-run partnership with Temba Bavuma swung the second test at the Basin Reserve, was sent to hospital for scans on the injury to his right index finger.
The scans did not show a fracture, team manager Dr Mohammed Moosajee told reporters in Hamilton, but the injury could rule him out of the IPL Twenty20 tournament to ensure he is fit for the one-day Champions Trophy and tour of England that follows.
"There is every likelihood he will miss the IPL because he will need at least four to six weeks recovery," Moosajee said.
"The risk is that if he doesn't have that four to six weeks rest, and he further aggravates it, it could jeopardise his participation in the Champions Trophy.
"With that in mind and with the England tour after that, it's important we give him the allocated time to recover for the injury."
The 24-year-old missed training on Wednesday and Thursday and Moosajee said a decision on whether he played in the third test would not be made until Friday.
Uncapped Heinrich Klaasen could take the gloves for the match at Seddon Park, which starts on Saturday.
South Africa lead 1-0 after their victory in Wellington and the rain forecast for the first four days of the Hamilton test could hamper New Zealand's hopes of saving the series.
(Reporting by Greg Stutchbury; Editing by Nick Mulvenney)
