"Code named 'Hurricane Piper Alpha', the objective is to smoke out war criminals of the northern Nigerian armed forces who have taken refuge in oil installations and give them instant jungle justice," the statement added.
A Chevron spokesman could not immediately confirm the incident, but said authorities are investigating the report, the AFP news agency reported.
'All out war'
It was not yet clear if any oil production in the country was affected because of the attack.
Mend declared an "all out war" against the Nigerian military last month after navy forces launched an attack on the group's fighters in Delta state. The group had made such declarations even in the past.
Violence in the delta has cut Nigeria's oil output by about a fifth since early 2006, causing foreign firms to remove all but essential employees and severely reducing the earnings of the country.
A number of armed groups operate across the region, launching attacks on oil installations and pipelines, as well as abducting oil workers and local officials.
Some are demanding greater autonomy for the impoverished Niger Delta or a greater share of oil revenues, while others are criminal gangs kidnapping for profit.