The latest deaths came the day after nearly 3,700 migrants were rescued at sea, according to the Italian coastguard.
The number rescued on Saturday was one of the highest recorded in a single day, raising fears that the tide of desperate people trying to reach Europe from Africa and the Middle East has not been slowed by recent disasters.
"Around ten people were found dead," the Italian coastguard said on Sunday, while several rescue teams worked in the waters off war-torn Libya, where most of the rickety, overcrowded boats set sail from.
Rescuers said most of the latest fatalities were on board distressed migrant boats which were intercepted. It was unclear how they had died.
Another three men drowned after they jumped into the sea, as they rushed to reach a coastguard vessel.
The Italian navy said its patrol ship Bettica picked up more than 400 migrants from two vessels on Sunday, among them some 60 women and around 15 children.
Meanwhile, the Libyan coastguard intercepted five boats with some 500 people on board, some eight nautical miles off the coast, and ordered them to head back for the city of Misrata east of the capital Tripoli.
Colonel Reda Issa told AFP that most of the migrants were Africans. He did not say what would happen to those intercepted, but Libya has a detention centre for migrants in Misrata.
Fleeing war, poverty
People smugglers have taken advantage of the chaos gripping Libya since the 2011 uprising toppled dictator Moamer Kadhafi.
On April 19th, some 750 migrants were killed when their trawler sank between Libya and southern Italy, sparking global outrage and demands for action.
Four days later EU leaders tripled the bloc's budget for patrols off Libya.
EU leaders are now seeking UN Security Council approval for military action against smugglers in chaos-ridden Libya. But rights groups have blasted the Europe for focusing on patrols rather than humanitarian efforts.
UN chief Ban Ki-moon has also urged the European Union to refrain from resorting to force.
On Saturday, Italian coastguards said 3,690 migrants were rescued.
This number wasn't far short of the record 3,791 migrants who were picked up on April 12th.
Video released by the coastguard showed people crammed onto a small boat. The migrants are later seen clambering aboard a rescue vessel.
Saturday's operations in the Mediterranean involved four Italian coastguard vessels, two Italian navy ships and two customs boats, as well as four cargo ships and tugs.
French patrol boat Commandant Birot, which was sent last week to boost the EU's Operation Triton patrols dealing with the influx of migrant boats, also picked up 219 people off the coast of Libya Saturday.
Most of the migrants rescued Saturday were being taken to Sicily or southern Italy, while some had already landed on the Italian island of Lampedusa.
However, two suspected people traffickers were to be handed over to police at the port of Crotone in Calabria in southern Italy.
Several hundred migrants, mostly Africans but also including many fleeing the civil war in Syria, set out from Libya every day, hoping to make it to Europe.
The number of migrants entering the EU illegally in 2014 almost tripled to 276,000, according to Frontex, nearly 220,000 of them arriving via the Mediterranean.
Some 1,750 migrants have died crossing the Mediterranean to Europe this year, 30 times more than during the same period in 2014, according to the International Organization for Migration.