As some of the rescued passengers arrived in Italy, Italian Prime Minister Matteo Renzi told reporters that four more bodies had been recovered. One man was reported dead on Sunday.
Renzi, speaking at a year-end media conference in Rome, said that rescue efforts should be completed within a "few hours" and praised the work of rescuers, who had helped avoid a "massacre".
The Norman Atlantic was carrying 478 passengers and crew and more than 200 vehicles.
Helicopter crews have been airlifting passengers from the upper deck of the ferry, now drifting in rough seas between Greece and Italy.
After initial rescue efforts were impeded by bad weather that stopped other ships getting close, Italian and Greek helicopter crews began the airlift on Sunday afternoon.
Rescuers worked through the night to pull people off the multideck ferry, the Italian navy said. Several passengers have been flown to Galatina in southern Italy.
A medical team and a flight operator had boarded the vessel to assist the passengers and crew as the rescue proceeds, a statement from the Italian navy said. Its San Giorgio amphibious transport ship is coordinating the rescue operation.
A merchant ship carrying a reported 49 of the ferry passengers, including four children, arrived in the southern Italian port of Bari on Monday, and Italian Admiral Giovanni di Tullio told Sky TG24 they would receive medical attention.
Bad weather hampered efforts overnight to attach cables to the ferry for towing, and a tug boat is expected to reach the ship to make another attempt on Monday, Greece's shipping minister Miltiadis Varvitsiotis told Skai TV.
No decision had been made on where the ferry would be taken, he said, although there had been expectations that it would be towed to the Italian port of Brindisi.
Eighty five people had been transferred to the San Giorgio by 0750 GMT and one person suffering from heart disease was taken to the Italian mainland by helicopter, the navy said.
The Italian-flagged ferry, chartered by Greek ferry operator Anek Lines, was sailing between Patros in western Greece to Ancona in Italy.
The cause of the fire has yet to be determined but the Greek coastguard said might have started in the parking area.

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