The crew members of the merchant tanker M/T Puli were abducted some 90 miles from the coast at around 1:30 a.m. (0130 GMT) on Monday, navy spokesman Chris Ezekobe said.
"Six crew members were abducted. They included the captain, the chief officer and chief engineer," Ezekobe said. "They were all Turkish."
The spokesman said the navy was going to board the vessel to speak to other crew members.
Last month, Nigeria and Equatorial Guinea agreed to establish combined patrols to bolster security in the Gulf of Guinea. The gulf is a significant source of oil, cocoa and metals for world markets, but pirates pose a threat to shipping companies.
They target oil tankers, usually seeking hostages for ransom and fuel to sell. Security analysts say the pirates have emerged from militant groups in Nigeria's oil-producing Niger Delta, such as the Movement for the Emancipation of the Niger Delta.
A lawyer for Kaptanoglu Group, an Istanbul-based shipping company, said the crew members abducted from the M/T Puli included the ship's captain and that those left behind were unharmed, according to the newspaper Hurriyet.
The tanker was carrying liquid chemical fuels and was traveling to Cameroon, Hurriyet said, citing the lawyer, Fehmi Ulgener.