The April 18th tragedy was the deadliest sinking in the sea between Europe and Africa in decades and sparked international calls for reinforced efforts to deal with the growing migrant crisis.
"The navy... has begun the recovery of bodies outside the trawler that sank on April 18th... from a depth of around 380 metres (1,246 feet)," the navy said on Twitter.
Only 28 people survived the shipwreck at the time, out of an estimated 800 on board when the vessel set sail from Libya.
Italian Prime Minister Matteo Renzi last month announced that he wanted the wreck to be brought to the surface and victims' bodies returned to their families for burial, saying the country could not "bury its conscience at a depth of 387 metres".
Also on Monday, Italian authorities said they were dealing with the arrival of another 4,400 migrants after a series of boats were rescued in the space of 48 hours from waters off Libya.
The rescues lifted to more than 69,000 - a record for the first half of the year - the number of migrants to have landed in Italy so far this year, according to figures compiled by the International Organisation for Migration (IOM).
The latest operations involved Italian ships, British, Irish and Spanish navy vessels and a boat operated by Malta-based humanitarian organisation MOAS, the Italian coastguard said.
The surge in migrant arrivals since the start of 2015 has left Italy's reception facilities, currently housing around 80,000 asylum-seekers and others seeking leave to remain in the country, under severe strain.
The situation is likely to get worse over the summer when the numbers of boats dispatched by people smugglers usually peaks. Of last year's arrivals, some 100,000 arrived between June and September.
The sharp increase in migrant arrivals has left Italy's centre-left government under constant attack from the country's vocal far-right and some regions are threatening to refuse to house any more migrants.
The crisis has also put Italy at loggerheads with its European Union partners over contested proposals to spread some asylum-seekers across the bloc and moves by neighbouring countries to tighten their border controls to restrict the numbers of migrants travelling out of Italy on their way to northern Europe.
Italy regards these moves as breaching the principles of solidarity and of free movement of people within Europe. On the other side of the debate there is a perception that Italy is not sufficiently rigorous about registering new arrivals at its ports and sending economic migrants back to their countries of origin as a deterrent to others thinking of making the trip.
According to aid groups, just over half of the migrants arriving in Italy have a legitimate claim to asylum, mostly as a result of having fled conflict in Syria or repression in Eritrea.
The civil war in Syria is also seen as being behind a surge in asylum-seekers arriving in Greece, where over 80,000 migrants have arrived this year, three quarters of them on boats which mostly leave from Turkey.
This year has also seen a sharp increase in the numbers of migrants dying during the crossings with over 1,800 fatalities reported en route to Italy and at least 27 off Greece, according to the IOM.