"During a search in the water near the central part of the ship, coast guard and police divers found remains which still have to be identified with DNA," the civil protection agency said in a statement.
The agency's head, Franco Gabrielli, told SkyTG24 news channel that he was sure they were human remains.
Thirty bodies were recovered from the wreck after the January 13th 2012 tragedy but two - Italian passenger Maria Grazia Trecarichi and Indian waiter Russel Rebello - are still officially reported missing.
The search for the bodies began earlier this week after the 114,500-ton ship was lifted upright last week in the biggest salvage operation of its kind.
Italian news agency ANSA said the remains that had been found were bones and that special permission was required from prosecutors to bring them to the surface.
The ship's captain, Francesco Schettino, is on trial for multiple manslaughter, abandoning ship before all the passengers had been evacuated and causing environmental damage.
The Costa Concordia crashed into a group of rocks just off the tiny island of Giglio with 4,229 people from 70 countries on board.