The controversy revolves around recent changes to immigration policy, particularly new directives governing visas, which have made it harder to bring a partner to New Zealand.
The new rules stipulate that couples must have lived together for 12 months in order to be granted a partnership visa. This change has hit the Indian diaspora hard, because of its tradition of arranged marriages.
Immigration officers were previously allowed to grant partnership visas at their discretion in exceptional cases, but new directives issued in May ended these exceptions.
Lees-Galloway said he has asked officials to review the decision.
"Whilst what Immigration New Zealand did was correct in law, it has left a lot of people worse off and I can understand the concern that people have expressed about this," he said in a video posted on Facebook on October 31.
"What I have done is asked Immigration New Zealand to look at their options and to see if we can find a fair and reasonable solution."
Indian organizations had spoken out against the changes, which they said felt like a punishment for the community.
"It's not like Kiwi culture where you live together for three, four or five years and then get married -- it doesn't work like that in India," Auckland Indian Association president Narendra Bhana told public broadcaster Radio New Zealand (RNZ) on October 23.
"I'm surprised that Immigration (NZ) has failed to understand that after all these years."
Peter Elms, national visa manager for Immigration NZ, said the move was made to prevent exceptions becoming the norm and contravening government policy.
"I don't believe this is a racist policy at all. It's a clear policy that's for all. The policies that we have in place for partnership are the policies for all non-New Zealanders," Elms told RNZ.
Controversy over the country's visa regime has been brewing for some time.
On October 23, Shane Jones, an MP for the nationalist-populist New Zealand First party and cabinet minister in the current coalition government, said people should leave the country if they don't agree with the rules.
"I would just say to the activists from the Indian community, tame down your rhetoric, you have no legitimate expectations in my view to bring your whole village to New Zealand and if you don't like it and you're threatening to go home -- catch the next flight home," he told RNZ.
The Waitakere Indian Association demanded an apology for what it called "insensitive remarks" from Jones.
In the 2018 census, 4.7% of New Zealand's population identified themselves as Indian, the fourth-largest ethnicity in the country after European, Maori and Chinese.
In addition, 2.5% of the population, or 117,348 people, was born in India.
As well as sparking accusations of racism against Indians, the controversy has exposed a rift between the center-left Labour party of Prime Minister Jacinda Ardern and its coalition partner New Zealand First.
Winston Peters, deputy prime minister and leader of New Zealand First, appeared to take credit for the stricter rules covering visas.
"Has New Zealand First had an influence on trying to tidy up the quality of information on which the immigration department relies? The answer is profoundly yes," Peters told RNZ.
However, Lees-Galloway said "there has been no government directive on partnership visas," RNZ reports.
New Zealand is set for a general election on or before November 21, 2020.