Workers for the salvage firm recovered the body from a cabin and handed it to the authorities to be identified.
They believe it is Indian waiter Russel Rebello, the last of 32 victims to be recovered from the wreck.
The cruise ship partially sank in January 2012 after it hit a reef off the Italian island of Giglio.
The Concordia was finally raised from the sea in July and towed to Genoa to be scrapped.
The captain, Francesco Schettino, is on trial for manslaughter and abandoning ship, charges he denies.
This is the third time local officials have said Mr Rebello's remains might have been found.
Tests on the other two occasions determined that the remains were not his.
This time, his brother Kevin said on Facebook he believed Russel was now coming home to India.
"I promised my family that I would bring back home my brother some day. When was the biggest question," he said.
"Finally, my younger brother's body has been found.
"Thanks to everyone who prayed and believed that one day he would be found."
Human remains found in the wreck of the Costa Concordia are believed to be the last victim of the cruise ship's 2012 capsize, Italian officials have said.