Nehra won the approval after his impressive effort in the second Hamilton Test, although India lost the match by four wickets earlier on Sunday to concede the two-test series 2-0.
The selectors had picked 14 players on Wednesday but decided to watch the performance in Hamilton before filling the final two slots.
The 24-year-old Patel has been rewarded for his consistency at domestic level in the last two seasons and has also been included among the 30
Fast bowler Javagal Srinath, leg-spinner Anil Kumble and batsman Yuvraj Singh will also join the rest of the squad in New Zealand.
Batsman Shiv Sundar Das, fast bowler Tinu Yohannan, left-arm spinner Murali Kartik and wicket-keeper Ajay Ratra will return home, a Board of Control for Cricket in India release said.
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