Earlier on Monday two Australian politicians spoke out against New Zealand's offer.
It is understood about 120 asylum seekers have left the facility since it was closed on October 31. Inmates sent out photos showing a line of buses and police vehicles outside the camp, built on a former PNG naval base, a day after Immigration Minister Petrus Thomas gave them 24 hours to get out.
Officials insist the transit centers are safe, and refugees would be offered the same level of care.
Some 400 refugees rebuffed continued efforts by Papua New Guinea authorities to convince them to move from a shuttered Australian detention camp Tuesday as the tense standoff over their future drags into a third week.
The PNG Supreme Court recently declared the Manus camp unconstitutional, forcing Australia to close the site.
They are barred from resettling in Australia and Canberra has struggled to transfer them to third countries.
"Australia must provide necessary [consular] services for those regretful refugees who wish to return to their countries", he concluded.
New Zealand Prime Minister Jacinda Ardern has blasted Australia's handling of the refugee crisis on Manus Island as unacceptable as she seeks another meeting with Malcolm Turnbull on the issue. I think it's clear that we don't see what's happening there as acceptable, that's why the offer's there'.
"We have grave concerns over the situation on Manus Island but also for the refugees on Nauru and that our hopes it to lend a hand as far as we are able in helping to resolve the situation", she said.
Canberra has been hopeful that up to 1,250 refugees on Manus and Nauru could be transferred to the United States under a resettlement deal struck a year ago.
Furthermore, New Zealand believes that US President Donald Trump is complicating the deal. Australia's offshore processing policy has been criticized by the United Nations and rights groups as essentially placing refugees in indefinite detention.