The French Government, while promising a "humanitarian welcome", says it will evaluate asylum requests on a case-by-case basis.
Prime Minister Lionel Jospin cautioned against granting them open-door asylum, saying that such a move would reward immigrant smugglers worldwide.
"We have to treat this kind of question at both a French level and in a European context in order not to give a sort of reward to those criminal elements involved in the traffic of men and women," Mr Jospin said.
The migrants, most of whom are said to be Iraqi Kurds, are being held at a disused French military base near Cannes.
Relations between the Kurds and their French hosts soured rapidly with some of the refugees staging a protest against their living conditions and urging the authorities to start asylum proceedings.
The brief demonstration prevented traffic from entering the camp in the southern town of Frejus.
Fierce row
A translator said they wanted to be classified as political refugees, and to be given warmer accommodation, more hot food and medical care for their children.
The arrival of the refugees has set off a fierce row among French politicians, already immersed in a nationwide municipal election campaign in which law and order is a major issue.
According to its own law, France cannot hold the refugees for more than 20 days.
If they cannot be expelled, they must be freed.
UK bound
The French media has been speculating that they would then head north to try to enter Britain.
Arrest warrants have been issued for the captain and crew, believed to belong to a Turkish and Iraqi criminal gang. They vanished after running the vessel aground.
Some 150 Red Cross staff work at the Frejus camp have been conducting medical examinations of the immigrants, who spent a week locked in the hold of the Cambodian-registered ship, East Sea.
The Kurds said they had paid $4,200 per adult to get to Europe, and $1,700 per child.