Local public prosecutor Salvatore Vella confirmed media reports that four people who survived the shipwreck told rescuers they were on a boat carrying 45 people, including three children.
Vella's office has opened an investigation into the incident.
The 7-meter-long boat set off on Thursday morning from Tunisia's Sfax, a hot spot in the migration crisis, but capsized and sank after a few hours when hit by a big wave, the survivors were quoted as saying by Italian news agency Ansa.
The Sea-Watch charity rescue group said one of its surveillance planes spotted them being rescued by a cargo ship. They were then transferred onto an Italian coast guard vessel and disembarked in Lampedusa, where they shared their story.
"They said they were among the few aboard (the sunken boat) with a life jacket, and (after the shipwreck) they remained in the water until they found another empty boat," Sea-Watch said in a statement.
The migrants arrived in Lampedusa exhausted and in a state of shock, and are due to be questioned by police, prosecutor Vella said. They are presumed to have spent several days adrift at sea with no food or drinking water.
The Italian coast guard did not respond to a request for comment.
A source with knowledge of the matter said it was unlikely that the shipwreck experienced by the survivors was one oftwo the coast guard had reported on Sunday.
At the time, the coast guard said they had rescued 57 people and recovered two bodies, amid media reports that at least one of the sunken boats had also set off from Sfax on Thursday.
Separately, Tunisian authorities said on Monday that they had recovered 11 bodies from a shipwreck near Sfax on Sunday, with 44 migrants still missing from that sinking.
Italy, a major route into Europe for hundreds of thousands of asylum seekers and other migrants, has seen an increase in migrant boats so far this year, with around 93,750 migrant arrivals by sea according to interior ministry data.
The figure compared with about 44,950 arrivals in the same period of 2022.
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