Thirty-nine migrants were rescued after the vessel sank on Friday afternoon some 40 nautical miles from the Libyan coast, the navy said, adding that the search for survivors continued on Saturday.
A frigate and a patrol boat deployed to the scene, backed up by a helicopter and a merchant ship.
An Italian tanker, which has picked up some 700 migrants from stricken vessels in recent days, along with the 10 bodies, is headed towards Sicily where it is expected early Sunday, the navy said.
Also Saturday, the Italian coastguard rescued some 300 migrants thought to be Syrians, including 93 children and six women, from a fishing boat near Calabria in southern Italy.
Thousands of people seeking asylum or a better life in Europe have died in recent years trying to cross the Mediterranean.
Warmer weather prompts a surge in migrants each year, and thousands of people - especially Syrians, as well as Eritreans and others from impoverished sub-Saharan countries, have landed on Italian shores, where processing centres are already saturated, notably in Sicily.
The Italian government says more than 50,000 people have arrived in Italy since the start of the year, equalling the total number for all of 2013. Some 2,200 have arrived in nearby Malta.