RNZI reports Papua New Guinea ordered the closure of the Manus camp after the country's Supreme Court ruled the facility unlawful, leaving the fate of the 850 people held there up in the air.
Australia and Papua New Guinea each claim each other is responsible for settling the hundreds held on Manus.
The injunction, filed in Australia's High Court on Wednesday, calls for the asylum seekers to be sent to Australia and not to Nauru, Matthew Byrnes, one of the lawyers acting on behalf of the majority of those held on Manus, told Reuters.
"We are hopeful that we will be successful with this filing," he said.
Australian Prime Minister Malcolm Turnbull maintains the asylum seekers will not be resettled in Australia.
The detention centre on Nauru houses about 500 people and has been widely criticised by the United Nations and human rights agencies for harsh conditions and reports of systemic child abuse. Many staying there have self-harmed.
A Somali woman on Monday set herself on fire at the camp, the second such attack in a week, with critics blaming Australia's policy.
"With two refugees setting themselves on fire after being sent to offshore detention camps, will (the Minister for Immigration) finally accept the consequences of putting desperate people in a situation even worse than the trauma they are fleeing?," said Adam Bandt, member of Parliament for the Australian Greens.
Manus detainee welcomes change in security
The Manus offshore processing facility is now being controlled by Papua New Guinea police rather than Wilson Security staff contracted by Australia's immigration department.
A Kurdish/Iranian journalist detained on Manus, Behrouz Bouchani, says since the PNG Supreme court ruled that detaining people against their will at the facility was illegal, restrictions at the camp have been relaxed.
He says detainees now have freedom of movement within the compound and are allowed to use mobile phones.
"Wilson Security, they are only watching and they dont have powers," he explained.
"PNG police now controlling the prison, which is very good because they are much better and they are kind with us, and they don't make any stupid rules, because Wilson Security, they were torturing us by lots of rules."