Act leader David Seymour will tomorrow present a petition, with almost 15,000 signatures, to Parliament asking for the family to be granted residency by special direction.
It is a similar story for Steve and Gail Webster of Nelson, who may be forced to leave in June after working and living in their community.
They moved from the United Kingdom in 2012 with their two teenage daughters.
The couple own Earthbloom Flower Shop in Nelson and Steve, also a volunteer firefighter, is a car salesman. At present he is helping fight the fire devastating parts of Nelson.
A friend and former volunteer firefighter Ken Mahon has started an online petition, with more than 34,000 signatures by this afternoon, to prevent the family from being forced to leave New Zealand.
Ardern said both cases had been dealt with by Immigration New Zealand (INZ) and she understood both cases were made under the entrepreneur visa category.
"There are certain criteria that are set down for categories like entrepreneurs, and that's the process that Immigration follows," she told reporters at her weekly post-Cabinet press conference today.
"One of the issues, as I understand, that's been set down with that category is that, to prevent individuals coming in and purchasing businesses as a path to residency, they've put some quite tight criteria in there, and they've [INZ] simply followed that."
"On face value, from cases, I can see how the public have a bit of a view … the way certain cases may fall," she said.
To qualify for permanent residency under the entrepreneur resident visa, people applying must have first have held a three-year entrepreneur work visa or other self-employment visa.
The criteria for the indefinite entrepreneur resident visa
• You must have been self-employed for at least 6 months
• If you're applying after less than 2 years of being self-employed, you'll need to have invested capital of at least NZ $500,000 and have created 3 new jobs in New Zealand.
The criteria for the three-year entrepreneur work visa
• You must make a capital investment of at least NZ$100,000. If your business is in the science or ICT sectors, or shows a high level of innovation or export potential, INZ may be able to waive this requirement
• You must score at least 120 points on the INZ Entrepreneur Work Visa points scale
* You must provide a business plan for the business you intend to buy or set up in New Zealand.