There are a number of children aboard the Oceanic Viking
Australian authorities have said 78 Sri Lankan asylum seekers in Indonesia will not be taken to Australia, their intended destination.
The ethnic Tamil asylum seekers have spent 10 days on an Australian customs ship in Indonesian waters.
Indonesia agreed last week to take the group to have their claims examined, but local officials are refusing to allow the Australian vessel to dock.
Australia has been faced with a sharp increase in asylum seekers this year.
"There's an agreement between Australia and the government of Indonesia that the people who were rescued in the open seas will go to Indonesia and be processed by the United Nations High Commission for Refugees in Indonesia," Australia's Foreign Minister Stephen Smith told ABC News.
On Tuesday, an Indonesian provincial governor stopped the Australian customs vessel Oceanic Viking from docking on Bintan island, near Singapore.
He said the island was not a "dumping ground" for refugees.
The Tamil asylum seekers were due to be taken to an Australian-funded detention centre on Bintan where other detainees have allegedly been beaten by guards.
The Tamils themselves have said they will not leave the ship voluntarily and have refused to co-operate with identity checks.
Mr Smith said they had no choice about where they claim asylum.
"They were picked up on the high seas. They were rescued on the high seas - it's not their choice."
The 78 Tamils are the latest in a flood of asylum seekers trying to reach Australia.
There has been an increase in asylum seekers reaching Australia by sea
Indonesia's navy also recently agreed to intercept another boat carrying about 250 Sri Lankans off the Java coast, who are currently refusing to leave their boat.
Human traffickers often use Indonesia or Malaysia as staging points before putting the migrants on boats for the hazardous voyage to Christmas Island, an Australian territory.
Australia receives just a fraction each year of what the UN estimates to be more than 15 million refugees globally, but the issue has split the country.
The opposition blames Prime Minister Kevin Rudd's government for relaxing the country's refugee laws.
Others criticise the government for not allowing the asylum seekers into Australia to have their cases examined.