Few refugees have accepted offers to resettle in Papua New Guinea and Cambodia because most hope that Australia will eventually take them in.
"Reports of a deal with the USA lack any detail whatsoever, but the idea of sending Muslim refugees to Donald Trump's US raises some concerns", Senator Nick McKim said in a statement.
Australian Prime Minister Malcolm Turnbull has confirmed that the deal to move refugees from his country to the United States is expected to be put into action next year, after the President-elect Donald Trump's inauguration in January.
US Secretary of State John Kerry, who is in New Zealand, confirmed the arrangement, adding that "we in the United States have agreed to consider referrals from UN refugee body UNHCR on refugees now residing in Nauru and in Papua New Guinea".
About 1600 people on Nauru and Manus Island would be eligible, but the numbers and assessment of people would be up to the United Nations High Commissioner for Refugees and the USA government, he said.
About 900 men are in camp on Manus Island while about 400 people are in the camp on Nauru with roughly the same number living in the community.
Immigration Minister Peter Dutton said asylum-seekers whose refugee applications are rejected would be expected to return home.
The need to resettle refugees has become more urgent, following Papua New Guinea's order to close the Australian-run detention center.
"We have a very long history of cooperation with the United States".
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Labor opposition leader Bill Shorten welcomed the potential "end to indefinite detention, especially for genuine refugees".
Prime Minister Malcolm Turnbull would not say whether he had discussed the deal with President-elect Donald Trump during their telephone conversation on Thursday.
The ABC reports that now there are 872 people in Australia's processing center on Papua New Guinea's Manus Island and 390 people in the Nauru center.
The director of legal advocacy at the Human Rights Law Centre (HRLC), Daniel Webb, said the third-party deal was a good start but there must be certainty for every person held in offshore detention.
"We must celebrate that these children are now finally being given the opportunity to fulfil their full potential".
Until late 2013, asylum seekers and migrants frequently attempted to reach Australia from Indonesia, usually by paying people smugglers for passage on boats that were often unseaworthy.
At a press conference in Canberra on Friday, Turnbull said the one-off deal would not apply to new asylum seekers making their way to Australia.
"There are options for people to go back to their countries of origin and if people have been found not to be refugees then this deal is not applicable to them", he said.
"Two-and-a-half months is plenty of time, and if it's the case it will be another great achievement from the Turnbull Government", he said.
According to News Corp Australia, the deal to resettle Nauru and Manus Island detainees is imminent following months of negotiations.