"I feel that this is in breach of their visa conditions ... it is clear that BOI(NZ) is bringing in overseas staff to take over local jobs given to New Zealand citizens," Mr Nair said.
"The actions of the ... bank are disadvantaging bona fide citizens of this country by putting us out of jobs for no fault of ours."
Mr Nair, from Fiji, has lived in Auckland for 25 years and is a citizen.
Bank of India New Zealand managing director Ranjitkumar Amarendra Jha said the bank was not aware of any complaint or investigation.
Mr Jha said the bank was not in the position to comment on this or on the personal circumstances of employees for privacy reasons.
Michael Carley, Immigration area manager, confirmed the agency had received a complaint and would "investigate as appropriate".
He said a specific purpose work visa was granted for an employee of the Bank of India to be seconded as branch manager in Auckland.
Under the scheme, those seconded to New Zealand by a multinational company can be issued a work visa for up to three years and no labour market test is required.
Last year, 144,978 work visas were granted to temporary migrant workers, a 5 per cent increase from the previous year.
The latest Household Labour Force survey has New Zealand's unemployment rate at 5.4 per cent.
What is a work visa?
Foreign nationals who do not have residence who want to work in NZ require a work visa. There are various policies that allow people to enter NZ to work, including essential skills, working holiday schemes and study to work.
How many work visa holders are there?
The number of migrants arriving for work over the last decade has grown continually. Last year, 144,978 work visas were issued, an increase of 5 per cent from the previous year
Who gets issued a work visa?
Immigration officers need to be satisfied that there are no suitable New Zealanders available to take up a job before a migrant worker is issued with a temporary work visa.
Why are MNCs different?
Employees seconded by multi-national companies can be issued a work visa for up to three years and no labour market test is required.