MELBOURNE, Australia — Refugees in one of Australia’s offshore detention centers who hope to be resettled in the United States have been offered another kind of resettlement — to a different offshore detention center.

Now that Washington has approved the first group of 54 refugees for resettlement, the Australian government is working to relocate refugees from its Manus Island facility in Papua New Guinea after that country’s Supreme Court ruled that the center was illegal.

On Tuesday night, the authorities in the Manus detention center posted a notice offering refugees voluntary relocation to another Australian offshore processing facility on Nauru, an island nation to the east. The Australian government has said that the Manus center will close by Oct. 31.

The question of where to place the refugees, all of whom are men, has been contentious. The Australian government had previously planned to transfer the men to the East Lorengau transit center, elsewhere on Manus Island. Refugees have insisted that East Lorengau was unsafe and that refugees there have been robbed and attacked with machetes.