Detained by Australia, the group of 22 comprised a Rohingyan man, his wife and 20 single Rohingyan, Afghan and Pakistani men.
The refugee advocate Ian Rintoul said one of the Rohingyan men has family in Australia but was told if he did not go to the US he would stay on Nauru for 20 years.
The advocate said he expected a total of about 130 refugees to leave Nauru for America this month.
About 130 refugees from Nauru and Papua New Guinea's Manus Island have so far been resettled in the US which agreed to take up to 1250 from Australian offshore detention.
Refugees from Iran, the largest national group in Nauru, protested last week at being excluded from the US deal, the advocate said.
Meanwhile, Mr Rintoul said three refugees were flown from Nauru to Taiwan last week for medical treatment.
Lawyers for refugees sent to Australia for treatment are often able to injunct their return to Nauru.