The Nigerian Maritime Administration and Safety Agency (NIMASA) said it had "commenced search (and) rescue operations" for the abducted crew from the MV Glarus.
The ship was attacked 45 nautical miles southwest of Bonny Island, near the southern city of Port Harcourt, as it traveled from Nigeria's commercial capital, Lagos, on Saturday.
NIMASA director-general Dakuku Peterside said the agency was working "to ensure that they are found and released unconditionally".
The search and rescue were being conducted with the Nigerian Navy and other security agencies, he told a news conference in Lagos.
"The issue of piracy in the Gulf of Guinea is a challenge we acknowledge and we are determined to tackle it head-on," he said, vowing "zero-tolerance" of the problem.
Kidnapping for ransom is rife across Nigeria, particularly in the oil-rich south where prominent individuals, members of their family and expatriates have regularly been seized.
Offshore, the Gulf of Guinea has become the world's piracy hotspot.
Out of the 16 incidents in which ships came under gunfire in 2017, seven were in the waters which stretch 5,700 kilometers from Senegal to Angola.
The Geneva-based company said it would not divulge the identities or the nationalities of the hostages for safety reasons but added it was in touch with their families.
But the Slovenian foreign ministry said on Monday that one of its nationals was among those taken.