"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".
This is a significant shift in approach but does not adjust Labor's stated policy in support of boat turn-backs and against Australian settlement for any asylum seeker who comes by boat.
"I've been meeting with those colleagues, as have the relevant ministers, and we have been acting on those issues", Mr Morrison said.
Mr Morrison said it was now up to the Opposition.
The chief medical officer, who looks after the health of asylum seekers and refugees on Nauru, has been kicked off the Pacific island.
Liberal MP Julia Banks and other MPs are supporting Mr Broadbent and Mr Laundy.
Under proposed laws being refloated by the government, any refugee settled in a third country would be banned from entering Australia, even under a tourist or business visa.
"I would urge them to reconsider their position on that".
Australian Prime Minister Scott Morrison said earlier this week he would allow people on Nauru to come to New Zealand if legislation was passed closing the back-door entry.
Doctors with direct knowledge of the health of the children on Nauru have also visited Canberra to tell MPs of the need for medical treatment in Australia, in conversations that have helped build pressure for a change in policy.
The ABC said it understood the Australian Government planned to call for a vote this week on the legislation.