Mr Brady is carrying out a wide-ranging inquiry into the service after being called in by the Government following a series of scandals and the resignation of its head, Mary Anne Thompson.
Mr Cosgrove today said fresh claims that immigration officers had been cutting corners when dealing with applications would be passed on to Mr Brady to include in his inquiry.
The claims were raised by National's immigration spokesman Lockwood Smith, who today released an anonymous letter from a person claiming to be an immigration officer.
The letter alleged officers were being placed under "threatening pressure" by managers to clear large volumes of applications, leading to shortcuts.
"Because it is far quicker and easier for staff to approve an application than it is to decline it, applicants from various high risk countries or dishonest applicants with high risk backgrounds are having their false jobs, false qualifications, false marriages all glossed over by immigration staff and getting their temporary permits all approved."
The officer also said it was rumoured that managers were being paid bonuses based on volumes of applications that were processed.
Dr Smith said the allegations were among the most serious ever made against a government department and called into question the security of New Zealand's border.
Public Service Association secretary Brenda Pilott said she had talked to at least two immigration officers who had said the same thing.
"They say staff have very hard targets to meet and that there aren't really enough staff to be able to do the job properly, so corners are being cut and immigration applications are being pushed through, perhaps without due consideration," she said on Radio New Zealand.
But Mr Cosgrove said the veracity of the anonymous letter was open to question, but it would be referred to Mr Brady.
He encouraged anyone else with concerns to also come forward.
"If other stakeholders in the community have more information they should bring it forward so the Auditor-General's inquiry can have it duly examined."
