Australia's Refugee Action Coalition says it has received information from Nauru that an Iranian asylum seeker attempted to commit suicide.
The group says the attempted suicide also highlights the lack of medical facilities.
Canberra resumed the transfer of its asylum seekers to Nauru last month, reviving its earlier policy of dealing with them abroad.
It has so far moved 211 asylum seekers to Nauru and plans to send several hundred others to its detention centre in Papua New Guinea.
According to the Refugee Action Coalition, the Australian Immigration Minister Chris Bowen has told the asylum seekers that it would be between eight months and a year before processing of their refugee applications would begin.