The callers demand that the victims pay money into a Western Union account or face serious consequences, such as deportation.
Calls have been made using numbers showing as the Immigration Contact Centre (09 914 4100) and Crime Stoppers (0800 555 111), but with an extra zero at the beginning.
The scammers are using a technology known as caller ID spoofing that allows a legitimate phone number to appear when the call is actually being made from another number, quite possibly from outside of New Zealand.
This particular scam has been going on since around mid-2013 and appears to target only Indian nationals.
Immigration New Zealand first posted a warning about the scam on its website on 20 October 2013 and Consumer Affairs last posted a warning in August 2014.
To date, almost 300 Indian nationals have reported being called by the scammers with the dollar values involved typically ranging from $1000 to $5000.
Immigration New Zealand has confirmed it never requests money over the phone.
Further advice can be found on the Immigration New Zealand and Consumer Affairs websites.