On Wednesday afternoon, Senator Lambie revealed she had demanded one condition from the government in exchange for her support for the repeal of the medevac laws. She did not outline the details of her ultimatum.
'Sensible, reasonable proposition'
The Senator has described the condition as a "sensible, reasonable proposition" that was developed after extensive consultation.
She said the request recognised the government's concern with "the way that medevac is functioning."
Sources who have been involved in Senator Lambie's negotiations say they outlined the option of third-country resettlement in New Zealand.
Nine Entertainment's Sydney Morning Herald reported Thursday morning that the move was proposed as a way of settling "as many" asylum seekers as possible who have been detained under Australia's offshore detention policy.
UN human rights chief urges Australia to keep medevac bill, slams 'harmful' offshore sites
Around 263 refugees and asylum seekers are detained on Nauru, with another 221 in Papua New Guinea.
A further 642 have been resettled in the US under a 2016 deal between former leaders Barack Obama and Malcolm Turnbull. Australia agreed to resettle Central American refugees in Costa Rica camps in exchange for the US resettling up to 1250 refugees detained on Nauru or Papua New Guinea.
In July, New Zealand Prime Minister Jacinda Ardern confirmed her country is still open to resettling 150 refugees.
The option was first raised by former New Zealand Prime Minister John Key in February 2013.
Home Affairs Minister Peter Dutton said in July said Australia would consider accepting the offer on the condition that it did not incentivise people to take part in people smuggling boat operations.
Medevac laws do not undermine border security: Lambie
Senator Lambie says she supports boat turnbacks.
"Boat turnbacks work. The promise that nobody who illegally comes by boat will ever be resettled in Australia is an important one," she said.
"I support the government's position on Operation Sovereign Borders. I do not believe this position is undermined by the presence of medevac."
More than 4500 doctors signed an open letter on Wednesday urging Ms Lambie to vote to save the medevac laws.
Government tight-lipped on negotiation with Senator Lambie
Minister Dutton said he wouldn't comment on Senator Lambie's condition.
“I think Jacqui can support the bill,” he told Sydney radio 2GB.
Labor leader Anthony Albanese says he supports revisiting the New Zealand resettlement option.
"Well, we think that medevac is working. And we think that the government, if it wants to resolve the issue, we think that it needs to work out third-party settlement options.
"There’s one that has very clearly been available for some time, in New Zealand. And that would resolve the issue because no one would need to be transferred to anywhere from Nauru or Manus because there wouldn’t be people on there."