The fishing vessel's crew, who were rescued by other Vietnamese ships after the Monday afternoon incident, said their boat was encircled by 40 Chinese vessels before being rammed, the official Vietnam News Agency reported.
"I confirm that the fishing vessel was rammed," an official with the Vietnam Regional Maritime Search and Rescue Coordination Centre in the central city of Danang told AFP.
The sinking occurred about 12 nautical miles southwest of the rig, the official added, requesting anonymity. "The 10 fishermen on board are on land now. They are safe. The ship sank."
It was the first ship reported sunk since the dispute flared in early May. The standoff has seen repeated skirmishes between dozens of Chinese and Vietnamese vessels, including many civilian and fishing boats. The confrontations have included reported rammings and the use of water cannon.
Relations between frequently testy communist neighbours Vietnam and China have plummeted over the oil rig's presence, which has exacerbated an increasingly heated dispute over territorial claims in the area. The oil rig is positioned in the vicinity of the contested Paracel Islands.
No official comment was seen from the Vietnamese or Chinese governments. But in Japan, government spokesman Yoshihide Suga said if the report was true, it was an "extremely dangerous act". "It's important that relevant countries abstain from unilateral actions that raise tensions and that the countries act cool-headedly, observing international laws," said Suga, the chief cabinet secretary.
Japan is engaged in a thorny maritime territorial dispute of its own with Beijing in the East China Sea.
Tensions over the oil rig sparked violent anti-Chinese riots in Vietnam. Beijing says four Chinese citizens died in the unrest, while Vietnam says three Chinese died. Hundreds of people have been detained over the riots, and two men on Sunday became the first sentenced to jail, receiving terms of one and three years.
Vietnamese Prime Minister Nguyen Tan Dung said last week that China's placement of the rig in the contested area had "seriously threatened peace". He said during a meeting in Manila with President Benigno Aquino of the Philippines – which also is facing off with China in disputed waters – that the two leaders agreed Beijing should be condemned by the international community for its behaviour in the South China Sea.
China claims nearly all of the sea, even waters approaching the coasts of its neighbours and has become increasingly assertive in staking those claims.
Vietnam, the Philippines, Brunei and Malaysia, as well as Taiwan, have competing claims to parts of the body of water.