Bayelsa state police say they learned of the ransom after capturing members of the criminal gang believed responsible.
Hyundai Heavy Industries has not commented on the reports.
The BBC’s Habiba Adamu, in Abuja, says it is rare for anyone to acknowledge ransoms paid to kidnappers in Nigeria.
The kidnapping of foreigners and wealthy Nigerians is common in the oil-rich Niger Delta region, and has become a lucrative trade for criminals.
Gunmen kidnapped the four South Korean and two Nigerian workers after storming a passenger boat on 17 December.
They were released days later but South Korean officials declined to say whether they had paid a ransom.
Bayelsa state police commissioner Kingsley Omire said most of the suspected kidnappers were later arrested, including the leader.
He told the police that a payment of 30m naira ($190,000) had been made, with each gang member getting 3m, Mr Omire told BBC Hausa.
The man is said to have later escaped from custody.
“The police did not know a ransom was paid before the release,” the state police chief said.
He said that paying ransoms would only encourage more kidnapping and urged people to instead contact the police, saying they have freed several hostages.
The mother of Finance Minister Ngozi Okonjo-Iweala was kidnapped in December and held for five days before being freed.
It is not clear whether a ransom was paid, or whether the motive was criminal or political.