Nigeria ceded Bakassi to Cameroon in August after a ruling by the International Court of Justice brought to an end a 15-year dispute over the peninsula, including rights to its oil fields and fishing grounds.
The handover was completed peacefully, but some local groups opposed the change of sovereignty and have threatened attacks.
The Reuters news agency reported the attack was claimed by the Niger Delta Defence and Security Council (NDDSC), a group active in the Bakassi peninsula and opposed to Cameroonian control there.
The NDDSC said it was working with a group called the Bakassi Freedom Fighters.
"I personally led the attack during which we seized 10 men whom we are holding as I am talking to you now." Reuters quoted Colonel Ebi Dari, a NDDSC commander, as saying.
"If the Cameroon government does not respond to our requests in three days' time, we will start killing them one by one," Dari said.
The armed groups are opposed to Cameroon's takeover of the Bakassi Peninsula earlier in the year and want the issue re-examined.
Stephanie Elbaz, a spokeswoman for Bourbon, said armed men in three speedboats attacked the Bourbon Sagitta while it was helping an oil tanker load crude oil at an offshore oilfield in the Gulf of Guinea.
She said the company was working closely with the French foreign ministry for the quick release of the hostages.
Heavily armed pirates have in the last year preyed on oil plants, oil and fishing boats and even coastal towns in the region.
Other Gulf of Guinea states that have seen similar attacks this year include Nigeria, Equatorial Guinea and Benin.