The border is closed to all but NZ citizens and residents with limited exceptions.
INZ resumed the processing of residence applications on May 14 - but said those wanting to enter the country on their resident visa for the first time will still be banned.
A woman in the US, who got her residence visa this year said she was told by INZ she could not join her husband because she had not travelled here as a NZ resident before the border closed.
"I was quite puzzled because I thought that because I hold a NZ resident visa, I should be allowed to enter if I could find a way to get there," she said.
The woman did not want to be named or pursue the matter further as she feared it could result in her visa getting revoked.
Meanwhile, temporary visa holders stuck overseas are also pleading with the Government to let them return to their homes in New Zealand.
When the Covid-19 pandemic reached New Zealand, the country went into lockdown and enforced border closures that did not allow those on temporary visas to return.
Jyotsna (37) an operations supervisor on a work visa, is desperate to return after being stuck in India for more than two months.
"My fear is that even when the lockdown is over, they won't let us in and it would be a mess for us," Jyotsna said.
Her work visa is valid until 2022 and her residency application under the skilled migrant category was lodged 11 months ago.
She has lived in New Zealand for the past five years and fears that if she loses her job, her visa will also get cancelled as it is linked to the job.
"If I am not able to return, I don't know how I am supposed to live. Funds are depleting. While I have been locked out of my only source of income, and I have little support here in India," she added.
INZ confirmed Jyotsna's application for residence was submitted in July last year, and was waiting to be allocated to an immigration officer for assessment.
"Miss Jyotsna has twice requested an exception to the border restrictions, however, as her circumstances were not of a critical humanitarian nature as per Immigration Instructions, she was not invited to apply for an exception," the INZ spokeswoman said.
She said travellers should seek approval from INZ before travelling.
Since the agency resumed processing visa applications, applicants in NZ are being prioritised over those who are offshore.
"People on temporary visas who are offshore are not able to come back to New Zealand unless they meet the strict border exception criteria," the INZ spokeswoman said.
"The bar for being granted an exception to the border restrictions is set high to help stop the spread of Covid-19 and protect the health of people already in New Zealand."
She said since the movement to Covid-19 alert level 3 on April 28, INZ has been able to process visas, including temporary visas for victims of domestic violence and online temporary categories such as Partnership visas, full fee-paying student visas and post study work visas.
Exceptions for entry may be granted to people who have a critical purpose of travel to NZ including essential workers, dependent children and partners of work or student visa holders who are in NZ and critical humanitarian reasons.
"Any decisions on when to lift the border restrictions will be made by the Government," she added.
