The Victorian Liberal Russell Broadbent has made a personal appeal to Bill Shorten to back government legislation stopping any asylum seekers resettled in New Zealand coming to Australia, declaring people languishing on Nauru need parliament to deliver a breakthrough.

Broadbent – one of four Liberals lobbying Scott Morrison for a circuit-breaker to get at-risk children and their parents out of offshore detention – told Guardian Australia that Shorten “could be an enabler in this if he chooses to be”.

“I call on Bill Shorten to do everything he can possibly do to move this issue forward, including supporting the government’s legislation,” Broadbent said on Wednesday. “He has a moral obligation to back up his sentiment with action.”

With pressure mounting in his own ranks, and with the critical Wentworth byelection in play, Morrison has this week put the prospect of asylum seekers being resettled in New Zealand on the table – but only if the government can secure the passage of legislation intended to ensure there is no backdoor entry to Australia.

Labor is resisting that proposal, saying the legislation imposing a lifetime ban on asylum seekers who arrive by boat from ever being allowed into Australia is too far-reaching. Some in the opposition, and on the crossbench, question Morrison’s bona fides, pointing out the legislation is not yet on the notice paper for debate.

The New Zealand foreign affairs minister, Winston Peters, also expressed concern about the impact of the legislation in the event a breakthrough in Australia was reached.

“We’re going to have to consider whether or not, as a result of our 2013 commitment [to offer to take 150 refugees from Nauru], we end up with people who are second-class citizens in New Zealand,” Peters said. “Do we, in our endeavour to be humanitarian about it, end up with a substandard level of citizenship, which is not what this country is about?”

There is some thought in Labor ranks about countenancing the government’s proposal, despite the firm no from the leadership that was communicated to refugee advocates in meetings on Wednesday.

Shorten, like Morrison, faces internal pressure for a breakthrough. The Tasmanian Labor senator Lisa Singh told the Senate on Wednesday: “We cannot just sit here and do nothing. We must act.

“No matter the different ways we can get to the same conclusion, I think we all know that the solution is that we bring the children out of the misery that they currently find themselves in.

“The sooner that this parliament resolves this issue, the better. Not just for those children but for humanity’s sake, and for our nation’s sake, because everyone is watching.

“They’re watching the fact that – in this government’s name, and in this country’s name – we are leaving these vulnerable children in a terrible and perilous situation; terrible in the sense that I hope nothing happens that might result in the end of their lives.”

The government is engaged in discussions with some of the Senate crossbench to see if there are the numbers to pass the proposal in the event Labor continues to oppose it.

Some crossbenchers are attempting to ascertain whether the government is prepared to accept amendments.

Stirling Griff of the Centre Alliance put in a call to the immigration minister David Coleman on Wednesday in an effort to discuss options, but he said he was not disposed to supporting the legislation as it was currently drafted.

“If the bill is identical to the 2016 bill which was a catch-all, meaning it would also apply to the US and other resettlements, we would likely reject it like last time because it is cruelty for cruelty’s sake,” Griff said. “The bill did not take into account a person’s future standing at the time they may decide to travel to Australia.”