The Refugee Action Coalition is pleading for a woman in the 37th week of her pregnancy to be flown from Nauru to Australia for the birth.
The Guardian reports the Kuwaiti refugee has previously had a miscarriage on Nauru, she has what is thought to be a benign tumour on the wall of her uterus and is also reported to have pre-eclampsia.
Officials at the Australian-run detention centre on Nauru said the woman was told she had to have her child on Nauru.
But the Coalition's Ian Rintoul said if Australia doesn't act quickly it could face a similar situation to that experienced last May when Somali refugee Naima had to be airlifted to Brisbane in emergency circumstances after having a caesarian on the island.
"The reality is that these kinds of deliveries cannot be done safely in Nauru.
"And that's an indictment of the Australian Government not providing facilities on Nauru, there not being the resources and the proper medical equipment, staff, so that Nauruan people can have deliveries safely, but at the moment that does not exist on Nauru and she needs to be brought to Australian immediately."