Under that deal, up to 1,250 asylum seekers could be sent to the United States and Australia will in turn accept refugees from Central America.
By Sunday afternoon, about 600 refugees and asylum seekers remained at the Manus Island site, which closed on October 31 following a PNG Supreme Court decision.
New Zealand's offer to accept 150 refugees from Manus Island and Nauru was waved away, with Turnbull saying his government had to first consider its refugee-swap deal with the US.
Those inside claim they will be attacked by PNG locals if they move to an alternative unfenced accommodation at nearby Lorengau.
Australia has an obligation to do so under global human rights law and the 1951 UN Refugee Convention, he said.
The remote Manus Island centre has been a key part of Australia's disputed immigration policy under which it refuses to allow asylum seekers arriving by boat to reach its shores, detaining them instead in PNG and Nauru in the South Pacific.
United Nations rights spokesman Rupert Colville told a news conference in Geneva on Friday about the "unfolding humanitarian emergency" in the Manus island centre, where asylum seekers have been reported digging wells to try to find water. "We did it on an understanding that Australia was looking for a place".
Immigration minister Peter Dutton has repeatedly said no refugees from Manus will be resettled in Australia.
The center on Manus Island was ruled "unconstitutional" by a court in PNG past year.
"But also we have urged Australia and PNG to de-escalate the situation, resume basic services - water, electricity, medical services as well".
"At the moment hundreds of naked men are lying around me", wrote Boochani.
In his retirement, the Grand Chief says he is at liberty today to speak about the miserable manner in which Australia is trying to offload the Manus Island Refugees.
New Zealand citizens are typically allowed to work and live in Australia, a pathway that Australia fears could be used by the refugees to move to Australia.
"Our situation is getting worse and we need your help", the other man said.
Ardern's relationship with Australia has been controversial since her election, with Foreign Minister Julie Bishop describing the 37-year-old New Zealand leader as "untrustworthy" after an MP from her own Labour Party delved into former deputy PM Barnaby Joyce's dual Kiwi citizenship status.
Ms Ardern acknowledged on Sunday that the situation was becoming acute but understood Mr Turnbull's position to first pursue the U.S. deal because it would help substantially more people than New Zealand's small offer could. "You can't keep political hostages".
In the first official phone call between Mr Turnbull and the United States president, Donald Trump referred to this agreement as "a stupid deal" that would "make me look terrible", according to the full transcript released by the Washington Post.