Fishermen in Indonesia's Aceh province say they have been told by officials not to rescue migrants from boats off the coast, even if they are drowning.
At least 700 Bangladeshis and Rohingyas from Myanmar were rescued off Aceh last week by locals, bringing the numbers in camps there to at least 1,500.
An army official said it would be illegal for any more of the migrants to come to shore.
All countries in the region have closed their borders to the migrants.
Thousands of people - mostly Rohingya Muslims fleeing persecution and poverty in Myanmar, but also Bangladeshis looking for work - are thought to be stranded out at sea.
Aid agencies say people on board the boats are severely malnourished, and should be offered immediate assistance. Survivors who have made it to shore say there have been deadly fights on board over food.
Analysis: Jonathan Head, BBC News, southern Thailand
It is hard to imagine any governments taking a more hard-hearted stance than those in Southeast Asia towards the migrant boats off their coasts.
Malaysia has blockaded its north-western sea border to stop them entering. Thailand has hurriedly repaired boats' engines and shooed them over its border, despite near starvation and illness on board. Now fishermen in Indonesia say they have been ordered not to pick up anyone, even if they are drowning. Why? They certainly fear a deluge of migrants if they open the floodgates.
They blame Myanmar for causing this crisis through its terrible treatment of Rohingyas. Myanmar refuses to accept responsibility.
But arguing over who is responsible for these migrants should not be the issue right now. Saving lives should be.
On Monday, some of the Acehnese people involved in last week's rescue said fishing boat operators were now being told by military officials not to carry out any more rescues.
Nobody wanted to speak on the record fearing they would be punished by the government, says the BBC's Martin Patience in Langsa, where the migrants are being cared for.
But one fisherman told the BBC that despite the warning they would continue rescuing people if they saw them drowning.
"They're human beings; we need to rescue them," he said.
Military spokesperson Fuad Basya said fishermen could deliver food, fuel and water to the boats, or help with repairs, but that bringing them to shore would constitute an illegal entry into Indonesia.
Meanwhile the mayor of Langsa has said the city has no budget for aid on this scale, and that it has received no help from Jakarta.
"In short, yes, we need some help, immediately, from our national government or any other institution, including NGOs, to take care of the Rohingyas who are stranded in our place," said Usman Abdullah.
The UN has called on all nations in the region to give aid and shelter to people in distress at sea.