The development in talks comes shortly after last weekend Amnesty International, supported by sporting legends Sonny Bill Williams and former socceroo Craig Foster, lobbied for the offer to be taken up while Australian Prime Minister Scott Morrison was meeting with New Zealand Prime Minister Jacinda Ardern in Queenstown.
Foster welcomed Andrews' comments on Twitter as "encouraging news", but said it wasn't enough.
"Those still imprisoned need certainty, however 240 offshore can be sent to NZ and 140 in APOD's [alternative places of detention] don't need to be locked up while waiting. End the pain. Let them out and let them go."
The "Medevac law" was a piece of legislation introduced following a series of court-ordered medical transfers for asylum seekers detained in Nauru.
The law, which operated for about nine months in 2019, meant that offshore asylum seekers could be transferred to Australia to receive medical treatment.
Almost 200 people were transferred from Manus Island and Nauru over this period, with another 1200 in some form of detention or on final departure bridging visas in Australia.
About 80 people are still on Nauru and 91 in Papua New Guinea, according to The Age.
Of those who came to mainland Australia, fewer than 30 have left Australia. None have returned to offshore detention, with all either taking up third-country resettlement or returning to their country of origin, The Age reported.
The New Zealand Government has a longstanding offer to resettle up to 150 refugees a year who have been rejected by the federal Australian government because they attempted to arrive by boat.
Former home affairs minister Peter Dutton previously argued accepting the offer would encourage boat arrivals because refugees might consider the country a "backdoor route" to Australia.
This offer has been repeatedly rejected consequently, particularly after a failed attempt to legislate for a lifetime visa ban to stop them ever returning to Australia.
Instead, Australia signed a deal with the United States to resettle up to 1250 refugees.
So far 940 people have been resettled in there, while another 258 refugees have been provisionally approved for resettlement but await travel.
Optimism has grown the full quota will be filled and potentially more after President Joe Biden increased the annual humanitarian intake to 62,500 last month, including 6000 people from east Asia, which includes Australia.
Andrews said she had made it "very clear" to New Zealand there wouldn't be a backdoor method for refugees to be able to return to Australia.
"We're working through those issues now and we will continue to do so, and of course we will continue to work with the United States about resettlement options there as well," Andrews told The Age.
Amnesty International's Graham Thom told The Age the news was surprising, but he hoped it indicated Australia was close to finding a solution for families, men, women and children who had been held in limbo for more than seven years.
"Hopefully, these comments from Karen Andrews is a change of heart from the federal government and a sign that common sense is going to prevail at last," he told The Age.
"We know they have been trying to get people off Nauru and PNG because it's just not sustainable and it has been costing them a fortune."
