This time it's an unidentified grey item with a blue border.
Johny Begue said on Sunday he found the piece about 5.30pm on Thursday local time and turned it into the gendarmerie on Friday morning.
A special gendarmerie air brigade in Saint Denis, the capital of the French island, confirmed it received the item.
Begue found a wing fragment known as a flaperon on July 29 that French investigators identified in September as part of the passenger jet that disappeared with 239 people on board on March 8, 2014.
Begue said unlike the flaperon, there were no barnacles on the latest item, which he said was square and estimated that it measured 40 by 40 centimetres.
"I was running. After, when I stopped to rest, that's when I found the piece" lying on the stony beach several metres from the water, Begue told The Associated Press by telephone.
"The same beach and nearly the same place."
He said the piece he found on the Saint-Andre beach was thinner and smaller than the flaperon, but the material had the same appearance, with a honeycombed interior.
"It looks like the other one, but I don't know if it's part of the plane or not. Experts will say," the 49-year-old Begue said.
The gendarmerie's Territorial Air Brigade confirmed that Begue turned over the piece on Friday morning, but had no further comment.
The flaperon, which had a serial number, was sent to the French Accident Investigation Bureau's research laboratory near Toulouse where it was positively identified as a part corresponding to the missing Malaysian plane, a Boeing 777.
Begue's latest discovery came just days after an American, Blaine Gibson, found an aeroplane part in Mozambique, also with a coastline on the Indian Ocean but west of Reunion.
Australian officials still expect to be sent the second potential piece of wreckage from MH370 for testing but are yet to receive any reports of the third find.
An Australian Transport Safety Bureau spokesman told AAP on Sunday they had not received any reports of a third find.
He said the second possible piece of MH370 debris, discovered on a sandbank in the Mozambique Channel last week and with the words "NO STEP" on it, was still expected to be sent to Australia for testing.
But it could go to Malaysian authorities instead, with the likely involvement of manufacturer Boeing, he said.
Tuesday marks two years since the Malaysia Airlines jetliner disappeared en route from Kuala Lumpur to Beijing, killing 239 people, including six Australians.
It's believed it crashed somewhere in the southern Indian Ocean, off Western Australia.