The relocation of the men is designed as a temporary measure, allowing the United States time to complete vetting of asylum seekers as part of a refugee swap deal, agreed on previous year, under which Australia will accept refugees from Central America.
"UNHCR, the UN Refugee Agency, today reiterates its call to Australia to stop a humanitarian emergency unfolding on Manus Island, Papua New Guinea", the organisation said in a statement.
New Zealand's Labor Prime Minister Jacinda Ardern is expected to urge her center-right Australian counterpart, Malcolm Turnbull, to let her country accept some of the refugees held on Manus Island and another Australian-run migrant camp on Nauru.
"We call on the Australian government. who interned the men in the first place to immediately provide protection, food, water and other basic services", United Nations rights spokesperson Rupert Colville told a news briefing.
Australia wants them to move to facilities in the island's main town, Lorengau, but the refugees fear more violent attacks by locals opposed to the relocation.
Amnesty International executive Director Grant Bayldon says while the rejection is disappointing, the focus needs to keep coming back on Australia to do the right thing.
Under its tough immigration policy, Canberra sends asylum seekers who try to reach Australia by boat to two camps, in Manus and Nauru, and they are barred from resettling in Australia.
United Nations rights spokesman Rupert Colville told a news conference in Geneva on Friday about the "unfolding humanitarian emergency" in the Manus island center, where asylum seekers have been reportedly digging wells to try to find water.
He further clarified that when he took office in 2002, there was already an arrangement in place on Manus Island for boat people who had fled their homeland destined for Australia.
"We repeat our overall concerns about Australian offshore processing centres which are unsustainable, inhumane and contradictory to its human rights obligations", Colville said.
Recorded messages from detainees on Manus Island have been played to a rally of hundreds of supporters in Melbourne's CBD.
Around 500 of the men have still not had their asylum claims processed, he said. "They force us to starve".
The government has been facing growing pressure to accept the offer in recent days, with Opposition Leader Bill Shorten saying on Friday that Mr Turnbull should honour New Zealand's proposal and Liberal MP Kevin Andrews breaking ranks to join that call on Sunday.
Ms Ardern said New Zealand's offer had not been rejected outright.
"I acknowledge that, while New Zealand has not had to contend with these issues on our shores, it's hard to ignore the human face of this situation and nor should it be ignored", wrote Ardern.
Despite this, Australia have placed priority on an agreement with the United States to settle 1250 refugees.
"Australia blocking but has no right to say no", he wrote on Twitter.
"We [PNG] can't all be blamed for resettling them here".