The group, who were part of the 157 asylum seekers detained at sea for weeks on a Customs ship, arrived at the detention centre yesterday evening, a Curtin spokesman said.
More asylum seekers are expected to arrive at Curtin today.
The Refugee Action Coalition says the asylum seekers were seen disembarking in the early afternoon on Sunday.
They were headed to the island's airport from where they were due to be flown to Curtin detention centre in Western Australia.
"There's no schedule, but given the lack of facilities on Cocos Island, they will probably try and get them on a plane soon - not everybody by today, but most," spokesman Ian Rintoul told AAP.
The federal government did not confirm the movements.
The group of 157, including children, were being detained at sea since their boat was intercepted 27km from Christmas Island on July 7.
After arriving at Curtin, they will undergo identity checks by Indian consular officials.
India has agreed to take back its citizens and will consider taking Sri Lankan nationals who are Indian residents.
Australian Greens say the group has a legal right to apply for asylum in Australia.
Greens senator Sarah Hanson-Young is travelling to Curtin in the next few days to inquire on the children's welfare and to brief the asylum seekers of their rights.
"Just because (Prime Minister) Tony Abbott wants to trade in their lives, doesn't make it legal," she told reporters in Adelaide on Sunday.
Mr Abbott has said the handling of the asylum seekers is not illegal and in accordance with Australia's international obligations.
But he stressed they could never call Australia home because they did not come the "right way".
The decision to bring the group to land pre-empts a High Court challenge against their detention at sea.
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