Flight MH370 was travelling from Kuala Lumpur to Beijing in March 2014 when it vanished with 239 people on board.
The developments are the closest yet into potentially explaining what happened to the aircraft and its crew and passengers.
Malaysia's prime minister Najib Razak said the debris - part of a wing - will be sent to the French mainland for investigation.
He said: " Initial reports suggest that the debris is very likely to be from a Boeing 777, but we need to verify whether it is from flight MH370. At this stage it is too early to speculate.
"To find out as fast as possible, the debris will be shipped by French authorities to Toulouse, site of the nearest office of the BEA, the French authority responsible for civil aviation accident investigations.
"A Malaysian team is on the way to Toulouse now. It includes senior representatives from the Ministry of Transport, the Department of Civil Aviation, the MH370 investigation team, and Malaysia Airlines.
"Simultaneously, a second Malaysian team is travelling to where the debris was found on Reunion.
"The location is consistent with the drift analysis provided to the Malaysian investigation team, which showed a route from the southern Indian Ocean to Africa.
"As soon as we have more information or any verification we will make it public. We have had many false alarms before, but for the sake of the families who have lost loved ones, and suffered such heartbreaking uncertainty, I pray that we will find out the truth so that they may have closure and peace."
British naval experts could yet help in the search for further remains, after David Cameron offered assistance to the Malaysian leader.
In talks with Mr Razak, Mr Cameron extended the offer of support.
A British official said: "We have some experts in the MoD who work in mapping, naval surveys and things like that.
"It's not a detailed offer at this stage. If they are interested we will follow up on it."
The source added: "We are offering brainpower not ships."
Dr Erik van Sebille, oceanographer at the Grantham Institute, Imperial College London, said current models mapping the ocean and currents support the theory that the object found on Reunion could be a piece of debris from the flight MH370 wreckage.
He said: "Starting from the spot where the object was found, we used computer simulations to retrace its path and determine where the aircraft might have entered the water 17 months ago.
"The chaotic nature of the ocean means that we can't track the exact location, but we can pin it down to an area a few hundred miles in diameter off the coast of north-western Australia. This location is consistent with the other evidence we have.
"The holy grail now is determining the location of the aircraft's black box, which is believed to be very close to the site of impact. The best way do to this would be to find more pieces of debris and then apply the same simulations to track their origin in the ocean."
Dr David Ferreira, an oceanographer at the University of Reading, said the strong currents of the Indian Ocean could conceivably have pushed debris from the flight several thousand 3-4,000km from the area where it was believed to have gone down.
He said: "This is on the fast side of the range of possibilities, but is still perfectly possible."
A French law enforcement helicopter is scouring the waters around Reunion in hope of spotting more debris like the piece of aircraft wing.
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