A defence ministry official said the warship was keeping a distance of about 50 kilometres from the South Korean tanker, which has a crew of five South Koreans and 19 Filipinos.
The tanker, carrying an undisclosed amount of crude oil, was seized Sunday on its way from Iraq to the US state of Louisiana.
The South Korean destroyer, which was in the Gulf of Aden on anti-piracy operations with about 300 soldiers on board, was ordered to move towards it.
A spokesperson for tanker owners Samho Shipping said there had been no word from either the crew or hijackers since Sunday, after the ship sent a distress call to the South Korean destroyer saying it had been boarded by three pirates.
The foreign ministry has said the government would not engage in any negotiations with the pirates and would leave it to the ship's owner.
Somali pirates, targeting one of the world's busiest maritime trade routes, raked in an estimated 60 million dollars (44.7 million euros) in ransoms last year.