The comments, which emerged in a video on social media, have been condemned by political, medical and community leaders and sparked urgent audits of patient care at the nurses' workplace in Sydney'south-west.
NSW Police have seized CCTV footage from Bankstown Hospital as part of their investigation.
In the video shared by Israeli influencer Max Veifer, one nurse appears to boast about sending Israeli patients to hell if they had to treat them while the other says "I won't treat them, I'll kill them." Both are wearing hospital uniforms.
Solicitor Mohamad Sakr, representing the male nurse, said in footage aired by multiple broadcasters on Wednesday evening that his client is trying to "make amends for what has happened".
"My client sends a very sincere apology to not only that individual but to the Jewish community as a whole," he added.
The nurses have been stood down.
Police will provide an update on their investigation on Thursday.
"This is a sad day for our country, it is unthinkable that we are confronted with, and forced to, investigate such an appalling incident," NSW Police Commissioner Karen Webb said.
"The speed in which this incident was reported by NSW Health significantly assisted detectives in what is a very serious investigation."
Detectives have examined CCTV footage, interviewed staff and pinpointed areas within the hospital where they believe the video was allegedly filmed as part of the ongoing Strike Force Pearl investigation.
An initial examination by NSW Health found no sign any Israeli patients had been affected.
NSW Health Minister Ryan Park said there would be a "conversation" with staff at Bankstown Hospital on Wednesday morning, "making sure they know that this type of behaviour is unacceptable".
"I can assure the Jewish community and all of the community that they should have trust and confidence to feel safe and get the care they need when they enter any one of our hospitals," he told Nine's Today show on Thursday.
"The vast majority of the 180,000 who work in our NSW health and hospital system do incredible work delivering the best care they can for their patients every day."
Federal Aged Care Minister Anika Wells agreed, saying Australia's health system was world-class.
"I want people ... to know we do not tolerate this ... and I don't want people to hesitate for a second when thinking about whether or not it is safe to approach our health system," she told Today.
The video was widely condemned by political leaders, including Prime Minister Anthony Albanese who said the footage was "sickening" and shameful and Mr Park.
The male and female nurses were captured using a website for random video chats when they spoke with Mr Veifer.
The male nurse, later identified as an Australian citizen and Afghan refugee with six years of nursing experience, falsely claimed he was a doctor and told the Israeli man he was "going to go to" hell.
Meanwhile, the NSW Nurses and Midwives' Association has scrapped plans for a Thursday rally outside the NSW Parliament for better pay and conditions.
The union will instead hold a "solidarity action" against hate speech in Martin Place in the CBD.
"The NSWNMA reiterates its condemnation of all forms of racism, bigotry and hatred, including acts of anti-Semitism and Islamophobia," it said in a statement.
