The bones are "presumably human", Francesca Maffini, a spokeswoman for the civil protection agency which is overseeing the search operation, told AFP.
"They were outside the ship, on a part of the seabed that was exposed when the ship was raised," she said.
But she said there could be no certainties without DNA identification, which could take "several days".
The search began on Tuesday after the 114,500-ton vessel was hoisted upright last week in the biggest ever salvage operation of a passenger ship after the January 13, 2012 tragedy on the tiny island of Giglio.
Trecarichi's husband, Elio Vincenzi, and Rebello's brother, Kevin Rebello, have visited the island several times since the incident and are warmly welcomed by the small fishing community when they come.
Vincenzi earlier this year laid a plaque at the bottom of the sea close to where the ship foundered.
The ANSA news agency said the bones were found close to where the fourth deck of the ship had been lying. The fourth deck is where the lifeboats were and where many of the other victims were found.
The ship's captain, Francesco Schettino, is currently on trial for multiple manslaughter, for abandoning ship before all the passengers had been evacuated and for causing extensive environmental damage.
Hearings are set to resume on October 7 with a long list of witnesses for the prosecution set to testify.
Four other crew members and an executive from ship owner Costa Crociere have already been convicted after negotiating plea bargains but do not have to go to prison under Italian law as their sentences are for under three years.
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