Officials said it was too dark to be certain the objects were from Flight MH370, which had 239 people on board.
A multinational team is searching for wreckage and ships will try to confirm the find after dawn.
Investigators are also checking CCTV footage of two passengers who were travelling on stolen passports.
Malaysian military officials said on Sunday that the plane may have turned back from its scheduled route shortly before vanishing from radar screens, further deepening the mystery surrounding its fate.
Relatives of the missing passengers have been told to prepare for the worst.
Flight MH730 left Kuala Lumpur, bound for Beijing, at 00:41 local time on Saturday. But radio contact was lost at 01:30, somewhere between Malaysia and Vietnam.
Late on Sunday, the Vietnamese authorities said possible debris from the plane had been spotted in the sea off south Vietnam.
“We received information from a Vietnamese plane saying that they found two broken objects, which seem like those of an aircraft, located about 50 miles to the south-west of Tho Chu Island,” an unnamed official from the National Committee for Search and Rescue told AFP news agency.
“As it is night they cannot fish them out for proper identification. They have located the position of the areas and flown back to the land,” he added.
The potential debris was in a similar area to a possible oil slick seen by Vietnamese navy planes on Saturday, but officials have cautioned that this too may be nothing to do with the disappearance of Flight MH370.
There are now 40 ships and 34 aircraft from nine different nations taking part in the search for the missing plane in the seas off Vietnam and Malaysia.
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