The Sukhoi Su-22 fighters crashed offshore from southern Binh Thuan province's coast during a diving practice, Thanh Nien newspaper said.
Vietnam conducts regular military training exercises along its lengthy coastline.
The People's Army of Vietnam uses the Russian-made fighter jets for patrols over the portion of the contested Spratly Islands that Vietnam controls.
The country is currently locked in a long-standing territorial dispute with neighbouring China over a number of islands in the South China Sea, which Hanoi calls the "East Sea".
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Tensions came to a head when Beijing moved an oil rig into waters claimed by Hanoi last year, triggering deadly anti-China riots in Vietnam.
China controls the Paracel Islands, after seizing them from what was then South Vietnam in a brief battle in 1974.
The Spratly Islands are also claimed in whole or in part by the Philippines, Malaysia, Brunei and Taiwan.