An Afghan refugee sent to Nauru by the Australian government remains on the Pacific island nearly three weeks after a doctor advised he needed to be urgently evacuated to treat a potentially fatal heart condition.

The 42-year-old man, known by the pseudonym Abdullah, was taken to the Nauru Hospital on August 29 after suffering severe pain in his chest and numbness in his left arm.

The Australian-funded asylum seeker processing centre on Nauru.

Abdullah said the pain had him writhing "like a snake" on his bed in his home, a converted shipping container, before he was taken to hospital by ambulance.

He was told he needed to be flown to Australia or Papua New Guinea for treatment, but discharged himself because there was no one else to care for his sons, aged 12 and 13.