Thirty-one babies already born to asylum seekers who were transferred from Nauru to Australia before December 4 will be allowed to remain in Australia, Immigration and Border Protection Minister Scott Morrison announced on Thursday.

They will have their protection claims assessed as part of the legacy caseload under a deal with Motoring Enthusiasts Party Senator Ricky Muir, whose vote secured the passage of new migration laws through the Senate this month.

In a statement, Mr Morrison said the agreement was a “special one-off arrangement” which will allow the babies and their immediate families to remain in Australia.

“This includes their mothers, fathers and siblings,” he said.

“That is, around 80 family members, all of whom are already in Australia having been transferred from Nauru, for the birth of their child.”

Senator Muir said the announcement should bring relief to some of the affected families.

"I still have concerns for the children and families that are currently on Nauru and Manus Island, " he said.

"I intend on discussing these concerns... in early 2015."

But the arrangement will not extend to expecting asylum seekers.

Mr Morrison said pregnant women who had been transferred to Australia but who had not given birth prior to Thursday would be returned to Nauru with their babies.

"Illegal maritime arrivals currently at and being processed at a Regional Processing Centre (RPC) must not think this decision gives them a ticket to Australia," he said.

"It does not. They will remain and be processed at the RPC."

Greens Senator Sarah Hanson-Young welcomed the announcement, but voiced concerns over those not included.

“It’s good that these babies will get to stay in Australia, for now, but I hold grave concerns for the pregnant women, children and families who will be left on Nauru as a result of this deal,” Senator Hanson-Young said.

“The other babies and children, who are on Nauru right now, should be brought to Australia immediately.”