The vessel moved in to save ten men and an unaccompanied minor from a sinking boat after trying to alert Libya's coastguard on Thursday morning. The migrants were from Pakistan and Ivory Coast.
The Aquarius said it had contacted Italy, Malta and Tunisia asking for help. Rome passed on the information to the Libyan coastguard, who asked them to hand over the migrants to one of their patrol vessels.
The crew of the Aquarius responded that they could not legally comply as Libyan ports were not considered places of safety, said SOS Méditerranée, a French NGO.
"The Aquarius 2 [the official name for the boat] has recovered about 10 people in Libyan search and rescue waters, a few miles from the mainland, but refused to cooperate with the coastguard in Tripoli," Italy's Interior Minister Matteo Salvini tweeted on Thursday.
"Now it's floating in the Mediterranean. I say it again and again: go wherever you want, but not to Italy."
Since Salvini closed the country's ports to migrants in June, the Aquarius has been forced to find new havens for the migrants it saves and has landed people in Spanish ports.
The ship was recently forced to lay up in the French Mediterranean port of Marseille after Gibraltar revoked its flag. It is now registered in Panama.
Earlier this month, the UN refugee agency UNHCR said that although the numbers attempting the crossing from north Africa to Europe had fallen sharply, the danger of dying during the crossing had sharply increased.
