Italy's coast guard rescued the migrants off the shores of Sicily after reports came in of dozens of people clinging desperately to the tuna cage out at sea.
Survivors later told rescue workers that seven people with them had drowned after they had tried to scrabble on board the trawler when the crew cut the line linking the boat with the cage.
Witnesses said the victims had been pushed back into the sea, Italian media said.
Rescuers pulled more than 250 people in several boats to safety on Sunday, as the number of people who landed in Italy over the weekend rose to over 900.
More than 100 men from Eritrea were taken to Lampedusa island off the Sicilian coast on Sunday, along with a further 55 people from a different boat, some of whom had jumped into the water at the sight of the coast guard.
On Saturday, some 690 immigrants were rescued from boats with engine trouble or dinghies that had begun taking on water, including a baby girl born during the crossing from Turkey.
Good weather has raised fears of increased numbers of migrants trying to reach Italy by sea.