The group is challenging laws passed following the Bondi Beach terror attack, which the government has used to criminalise joining, supporting, fundraising or recruiting for White Australia.
At a High Court hearing on Wednesday, lawyers for the organisation pushed for the hate listing to be put on ice, at least until the full case could be heard in September.
A decision on the injunction would be handed down at 2pm on Thursday, Chief Justice Stephen Gageler said.
"Unless restrained, there will be an extinction of the organisation," White Australia's lawyer Peter King told the court on Wednesday.
Allowing the neo-Nazi group to be designated a hate organisation would render the broader legal question about the hate laws moot, because White Australia, which had brought the case, would no longer exist, Mr King said.
Mr King also said the Victorian corporations registrar had sent a "show cause" notice which meant that because of the anti-hate laws, White Australia would effectively cease to exist on June 25.
Commonwealth lawyer Brendan Lim SC said the party had other options, including asking for its registration to remain in place until the High Court battle is resolved.
Mr Lim argued granting the injunction would set a precedent that could have consequences far beyond the White Australia Party.
As a result of overturning the current law, "persons may apprehend that they can commit crimes with no risk of punishment or detection", he told the court.
He also warned the risks of allowing White Australia to continue in its current form, including the potential for its rhetoric to incite violence, far outweighed the harms of listing the organisation as a hate group.
In a video posted to encrypted messaging site Telegram on Tuesday night, notorious neo-Nazi Thomas Sewell said the move to ban his organisation breached the constitution's implied right to freedom of political communication.
White Australia is also seeking to register as a political party.
Home Affairs Minister Tony Burke declared the neo-Nazi organisation a prohibited hate group in May after receiving advice from spy agency ASIO.
White Australia's predecessor, the National Socialist Network, announced it was disbanding when the hate laws were introduced.
But in reality, Mr Burke said, the group had "phoenixed" - changing its name to White Australia and continuing operations with largely the same members.
Under the government's declaration, it is now a crime to support, fund or join the group.
Islamist organisation Hizb ut-Tahrir has also been banned under the post-Bondi laws.
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