The huge ship was carrying more than 4,000 holidaymakers and crew when it capsized after striking rocks on January 13, 2012, killing 32 people, including two whose bodies were not recovered.
The head of the civil protection agency, Franco Gabrielli, told reporters the remains discovered were "absolutely consistent" with the two missing people - an Indian man and an Italian woman.
However, their identities could be confirmed only after DNA testing, he said.
After lying on its side in shallow water ever since capsizing, the Costa Concordia was hauled upright last week in a complicated 19-hour salvage operation.
Recovering the human remains after 20 months under the weight of the 114,500 tonne vessel was "almost a miracle," Gabrielli said. A spokeswmoman for the civil protection agency all that was left of the bodies were bone fragments.
The ship is due to be towed away from the Mediterranean holiday island, probably by next spring, and eventually broken up into scrap.
