A wing part was also found on July 29.
The object, believed to be the door of an aircraft, was discovered just south of the city of St Denis.
It is said to have writing on it and possibly some illustration.
The Malaysia Airlines plane from Kuala Lumpur to Beijing disappeared in March 2014 with 239 people on board.
The first piece of wreckage has gone to France, where investigators in the city of Toulouse will seek to establish if it came from MH370.
It is believed to be a wing flap from a Boeing 777 – the same type of aircraft as the one that vanished.
Fragments of a suitcase found on the same beach in St Andre are also to be examined.
Investigators will begin work on August 5.
An Australian-led search effort for the plane has been focused on a vast area of the southern Indian Ocean about 2,500 miles to the east of Reunion.
Authorities believe MH370 veered off course on the way to Beijing and crashed into the sea – but they do not know why.
The estimated last location of missing flight MH370 was based on “pings” sent from the aircraft that were detected by satellite.
However, no physical trace has been found.
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