Australians want the government to accept an offer from New Zealand to resettle refugees in detention on Nauru, according to a new poll.

The YouGov Galaxy poll commissioned by the Sunday Telegraph reveals 80% of people want children and their families transferred off the island.

Support for the NZ deal was strongest among millennials, according to the poll of 1,027 voters across Australia by the News Corp paper.

Half believe it would not open the floodgates to more refugees while a quarter said they would be more likely to vote for the Coalition if it agreed to the resettlement deal. But 44% said it would not influence their vote.

The poll adds pressure to the Coalition after a week in which the opposition, crossbench MPs, a former home affairs department official, and Liberal MP Julia Banks urged the government to use the New Zealand deal to remove refugees and asylum seekers from Nauru.

On Tuesday Labor offered conditional bipartisan support to pass a bill to prevent refugees settled in New Zealand to permanently migrate to Australia but Scott Morrison repudiated the offer declaring “you don’t horse-trade on border protection”.

On Saturday, more than 1,000 people stopped traffic in the heart of Sydney while about 500 protesters in Melbourne rallied against the federal government’s offshore detention centres.

The health of asylum seekers on Nauru has shot to prominence in the past few weeks, with Scott Morrison under immense pressure to bring sick children and their families to Australia.

Eleven children were evacuated earlier this week, leaving 52 on the island. Refugee advocates say another four kids have since left Nauru.

Morrison said about 30 people had been taken off Nauru over the past few weeks.

“Those numbers have been coming down and we will continue to work on that,” Morrison told reporters in Sydney.

“We have just been getting on and doing it like a responsible and compassionate government should.”

Federal Labor MP Anthony Albanese said ahead of Saturday’s rallies that Australia could have strong borders “without losing our national soul”.

On Sunday independent MP Andrew Wilkie told the ABC’s Insiders that “many Australians see the solution as actually shutting down the centres and bringing everyone to Australia to be processed, or to have permanent refuge”.

“But if we’re not going to do that, then the government supported by the opposition has to at least get the sick children off Nauru,” he said.

Wilkie said that up until a week ago medical transfers from Nauru only occurred with a federal court order or the threat of one. He expressed hope that the most recent group brought willingly by the Australian Border Force “might signal a very significant change in government policy”.

Centre Alliance MP Rebekha Sharkie said that there has been “a slight move in government policy quite recently by not fighting those court orders and by actually uplifting children quickly”.

“But we still have, to our knowledge, 50 children on Nauru and we would say every day, is a day too long,” she said.