The fire broke out on the lower car deck of the Italian-flagged Norman Atlantic, which was travelling from the Greek port of Patras to Ancona in Italy, just before 6am local time (4am GMT).
Authorities confirmed there were 423 passengers and 55 crew members on board and said more than half of them remain trapped on the ferry as the rescue operation is being hampered by choppy seas and bad weather.
Some passengers have reportedly been stranded on an upper deck of the stricken ferry for more than six hours, as they are unable to be lowered to lifeboats due to the high seas.
Sofoklis Styliaras, a Greek passenger still on board the flaming vessel, told Mega television: 'The fire is still burning. On the lower deck, where the lifeboats are, our shoes were starting to melt from the heat. There's nowhere else for us to go. It's impossible to walk on the lower deck because of the heat.'
Another passenger told Greek TV: 'They tried to lower some boats, but not all of us could get in. There is no coordination. It's dark, the bottom of the vessel is on fire.
We are on the bridge, we can see a boat approaching... we opened some boxes and got some life vests, we are trying to save ourselves.'
Another, quoted by The Daily, appealed for help saying: 'We urgently need help. We cannot leave the ship. There are boats but we are trapped.'
Miltiadis Varvitsiotis, merchant marine minister, said a lifeboat carrying about 150 passengers had been lowered into the water but only 35 people had been moved to a nearby ferry so far because of the difficult weather conditions. The rest remain on the burning vessel.
He said the rescue operation had been made increasingly difficult by extremely poor weather conditions.
'This is a very difficult, a very complex rescue operation,' he said. 'The visibility is poor and the weather conditions are difficult.
'We are in constant contact with Italian authorities and the Greek armed forces. We are committed to rescuing everyone on the ship, and are trying to ensure that nobody will be left unaided.'
The ship, which was also carrying 222 vehicles, is not thought to be in any immediate danger of sinking.
It is currently about 42 nautical miles (48 miles, 78 kilometres) northwest of Corfu, close to the Albanian city of Vlora, but continues to drift towards Italian waters.
Several explosions have been heard from the lower deck of the ferry and there have been reports of rescue workers crying as they try to free those on board.
Authorities said there had been no reports of injuries from anyone on board.
Greek government officials said Prime Minister Antonis Samaras was in contact with his Italian counterpart, Matteo Renzi, to coordinate the operation 'at the highest level'.
Marco Di Milla, Italian coastguard spokesman, said the rescue operation was likely to last for hours.
He said it would be up to the Greeks, in command of rescue operations, to decide where the rescued passengers and crew would be taken.
Greek authorities have sent five helicopters and a military transport plane to the area to assist in the operation.
Italy has also sent resources to help with the rescue effort, including a coastguard boat, an Italian Navy helicopter and an Air Force aircraft.
At least eight merchant ships, which were in the area at the time of the fire breaking out, have also joined the rescue effort and are being used to form a barrier against winds of up to 55 miles per hour.
It is believed rescue crews are waiting for a tug boat to arrive to help tow the ferry to the nearest port.
