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."

The Tunisian interior ministry said the three, aged between 18 and 27, were members of a "terrorist cell", and that they were detained overnight.
Tunisian-born Amri, 24, was shot dead by police near the Italian city of Milan in the early hours of Friday.
Monday's lorry attack on the market left 12 people dead and 49 injured.
The interior ministry statement said Amri's nephew - the son of his sister - had confessed that he had communicated with his uncle via the encrypted chat application Telegram to evade security surveillance.
It said the three-member cell had been active in the towns of Fouchana, outside Tunis, and Oueslatia near Amri's hometown of Kairouan, about 150km (95 miles) south of the capital.
The statement added that Amri had sent money to his nephew to travel to Germany and join a jihadist group, and encouraged him to pledge allegiance to the so-called Islamic State (IS) group.
Meanwhile, intelligence services in Spain are investigating a possible internet communication between Amri and a Spanish resident on 19 December, Interior Minister Juan Ignacio Zoido told radio station COPE.
Tunisia has been under pressure to reform its security sector since it suffered from a spate of deadly attacks by IS militants in 2015.
With the help of countries including Germany and the UK, counter-terrorism operations and border security management have significantly improved, and the country has not witnessed a major terrorist attack this year.
But its nationals have been involved in some of the deadliest attacks to hit Europe this year. The attacks in Nice, and more recently in Berlin, were carried out by Tunisians who had moved to Europe long before IS existed, and neither appeared to have any Islamist ties before they left their homeland.
The latest arrests show that even if its nationals get radicalised abroad, that too can contribute to others joining their ranks from here. This is still a country that is contributing the largest number of militants in the region, and authorities here have not figured out how to put a stop to it.
Today, it is also facing the prospect of thousands trying to come back here as IS loses ground in some of its biggest strongholds in the region. People worry that the state is ill-prepared to deal with that influx, and there is concern over the threat that this looming reality poses domestically.
On Friday, IS released a video showing Amri pledging allegiance to its leader Abu-Bakr al-Baghdadi.
Amri was shot dead after opening fire on police officers during a routine police check in the Milan suburb of Sesto San Giovanni, after a three-day Europe-wide manhunt.
In the summer of 2015, a United Nations report said an estimated 5,500 Tunisians - mostly young people between the ages of 18 and 35 - were fighting in the ranks of terrorist organisations in Libya, Iraq, Syria and, to a lesser extent, Mali.
In November this year, the ministry of the interior in Tunis said about 800 fighters had returned to the country.
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.
[NEXT_CONCEPT]
Tunisian security forces have arrested the nephew of the Berlin market attacker Anis Amri and two other suspects, officials say.