The ABC said it understood the Australian Govenrment planned to call for a vote this week on the legislation.
The legislation would prevent asylum-seekers who arrived by boat after mid-2013 from applying for an Australian visa.
Backbenchers Russell Broadbent and Craig Laundy asked Morrison last month to evacuate all children and their families from Nauru.
Fellow Liberal MP Julia Banks was also reported as saying the situation on Nauru had reached a "tipping point".
New Zealand has a long-standing offer to take 150 detained refugees, an offer that has so far been rebuffed by Australia, which fears they may use New Zealand as a back-door entry to Australia.
A spokesman for Prime Minister Jacinda Ardern said today the New Zealand Government would wait to see the outcome of the Senate vote, but the offer remained the same.
Close to 6000 doctors have written to Morrison demanding the Government remove the 80 children from Nauru because of serious mental and physical health concerns.
"The medical situation is urgent for this vulnerable group of children," it reads.
"Medical experts, including doctors who have worked on Nauru, have spoken repeatedly about their concerns. They report widespread, significant deteriorations in physical and mental health.
"We call on you to show your commitment to the strong health care system we have in Australia, which we are the guardians of, by accepting the recommendations of many medical experts and bringing these children and families to Australia so their health care needs can be met adequately, and with appropriate clinical oversight."
Ardern raised the issue with Nauru President Baron Waqa when she was on Nauru for the Pacific Islands Forum last month.
She met people working with refugees in Nauru but not any of the refugees themselves during her brief visit to the island.
A spokesman for Prime Minister Jacinda Ardern said today the New Zealand Government would wait to see the outcome of the Senate vote, but the offer remained the same.