With a long coast facing the South China Sea, the Southeast Asian country is prone to tropical storms that can cause large numbers of casualties and serious damage to industrial production and properties.
The depression is forecast to strengthen into a storm in the coming hours, the government said in a statement, as it sent a warning to coastal provinces.
The storm would trigger heavy rains in central Vietnam, it said, adding that "the depression is evolving in a complicated manner, with possible changes in course, speed and strength."
Vietnam has been reeling from the impacts of typhoon Yagi, the strongest storm to hit Asia this year, which made landfall in its northeastern coast 10 days ago.
The typhoon and its subsequent floods and landslides have killed 298 people overall, with 35 others missing and nearly 2,000 people injured, the disaster management agency said on Wednesday.
Industrial production in several northern provinces was halted, 235,000 houses were damaged and more than 300,000 hectares of rice and cash crops were inundated.
The possible new storm would approach the central city of Da Nang late on Thursday or early Friday, according to the weather agency.
Authorities in Da Nang and neighboring Quang Nam Province, home to tourist town Hoi An, have warned vessels at sea about the incoming depression and readied search and rescue plans, according to their disaster management departments.