The Briton was the first expatriate oil worker to be killed after being taken hostage in Nigeria, Nigerian security sources told Reuters news agency.
Security sources said a firefight broke out between Nigerian forces and the kidnappers after they ran into a military patrol while the hostages were in their boats.
ENI closed down its Okono/Okpoho oilfield - which produces 50,000 barrels a day - after the kidnapping, an industry source said.
Another source said the attack was "community-related", referring to disputes between oil companies and nearby villages about employment, infrastructure and other benefits.
Dozens of mostly foreign oil workers have been seized this year but most have been released unharmed after talks involving the local government.
Nigeria - Africa's top oil producer - has already cut output by a fifth since February, when armed groups fighting for control of the delta's oil wealth attacked pipelines, platforms and export terminals.
The groups want to see the region receive a more money from the oil extraction that has resulted in huge revenues for the federal government and oil firms.