Hundreds of men have refused to leave the Manus Island camp in an increasingly tense stand-off with authorities since Australia declared the facility closed on October 31 and shut off electricity and water.
Ardern said she raised the issue with Turnbull in passing, but they have not discussed it further.
Immigration officials have returned to the Manus Island detention centre to destroy water supplies and makeshift shelters, increasing pressure on the men inside to leave.
"We need an urgent solution to it, whether legal or political". No matter what label you put on it there is absolute need and there is harm being done, ' she said on Sunday.
On Monday evening, refugee activists in Sydney took over the stage at a Wesley Mission human rights lecture - featuring guest speaker ex-Immigration Minister Philip Ruddock - to hold up banners saying "Evacuate Manus Now" and '#manus is a human rights crisis' and to call for the Australian government to act on the ongoing humanitarian crisis on Manus Island.More news: Philippines' Duterte set for drug war backing at summit
He said giving asylum seekers New Zealand residence could give them a back-door into Australia.
'I see the human face of this and I see the need and the role New Zealand needs to play. "I think it's clear that we don't think what's happening there as acceptable, that's why the offer's there", she said.
The renewed push comes close to two weeks after the detention centre officially closed and as another deadline for asylum seekers and refugees still inside to leave passes.
'It continues to be the easiest option is to go through Australia given the screening that's already occurred of those refugees.
Asked about criticism of her approach from National foreign affairs spokesman Gerry Brownlee, Ardern said she was happy with how it had been handled.
"Philip Ruddock was one of the architects of Australia's brutal and inhuman offshore detention regime".
According to an Australian journalist, Ardern and Turnbull "chatted informally" early Monday morning (NZT) following her remarks, although it was unclear whether Manus had been mentioned.