The 41 survivors say they were transferred to another vessel when it sank in the middle of the night.
They said that up to 500 people died, but coastguards in the region have been unable to confirm their accounts.
Numbers of migrants making the dangerous sea route f-rom Libya to Italy have surged this year.
The survivors, f-rom Ethiopia, Somalia, Sudan and Egypt, spoke to the BBC f-rom the southern Greek city of Kalamata, whe-re they are being held after their rescue.
According to the group, about 240 migrants left the Libyan port city of Tobruk heading for Italy.
Once out in the Mediterranean, they said they were transferred to a larger boat already packed with more than 300 people, which then capsized.
The survivors were then picked up by a cargo ship, whose crew told the BBC that the migrants initially refused to be handed over to the Greek coastguard as they were determined to get to Italy.
A Somali woman living in Egypt told the BBC Somali service that three of her relatives, whom she had not heard f-rom since they set out for Europe on Thursday, had died.
The presidents of both Somalia and the self-declared Republic of Somaliland offered their condolences over the incident. The Somali embassy in Cairo put the death toll at almost 400.
But the UN refugee agency (UNHCR) has cast doubt, tweeting that the information hundreds had died appeared "inaccurate".
The fact that the boat capsized at night in open sea may well have contributed to the lack of clear information available, correspondents say.
March migrant crossings up on 2015
9,600
migrants crossed the Mediterranean to Italy in March 2016
26,460
arrived in Greece f-rom Turkey
-
In March 2015:
-
2,283 made the sea crossing to Italy
-
7,875 made journey f-rom Turkey to Greece
In a separate incident, six bodies were recovered and 108 migrants rescued when a rubber dinghy sank off the coast of Libya, according to the organisation SOS Mediterranean.
The boat was partially deflated, taking on water and its engine was out of use, the rescue group said.
The number of migrants arriving in Italy f-rom Libya has surged recently - about 6,000 made the journey alone over a three-day period last week, the International Organisation for Migration said.
The deaths come on the eve of the first anniversary of the sinking of a migrant boat in the waters between Libya and Lampedusa in which up to 800 people may have drowned.
About 180,000 people have attempted to reach Europe by sea this year, with nearly 800 lives lost, the UN says.
