The left-arm paceman, 30, suffered the injury while fielding a ball off his own bowling during a match for the Otago Volts.
"The initial assessment is that Neil will require four to six weeks for recovery, but we'll be in a position to make a better judgement on that in the next couple of weeks," coach Mike Hesson said in a New Zealand Cricket statement.
"Neil's an important member of our test side and he is obviously desperate to play in such a big series, so we'll do everything we can to get him back on the park as soon as possible."
The three-match series begins in Dunedin on March 8.
Meanwhile, New Zealand added leg-spinner Ish Sodhi to their squad for the third and final one-day international against Australia on Sunday.
Opening batsman Martin Guptill is doubtful for the match due to concerns over a hamstring injury that saw him overlooked for selection for the second ODI in Napier on Thursday, which was abandoned due to an unsafe outfield.
"Martin has made progress in the last couple of days but it's not yet clear whether he'll be available for selection," New Zealand selector Gavin Larsen said. "At this stage it's probably 50-50."
The hosts lead the series 1-0.
(Reporting by Sudipto Ganguly in Mumbai; Editing by John O'Brien)
