The Movement for the Emancipation of the Niger Delta (MEND) said in an emailed statement it had struck the Shell Forcados platform in the Delta state. There was no immediate independent confirmation.
The report followed an announcement on Friday by four Nigeria militant factions to accept in principle an amnesty offer from the country's president, raising hopes Africa's top oil producer would halt a battle with rebels.
U.S. crude for August delivery was up 10 cents at $69.26 a barrel by 0750 GMT (3:50 a.m. EDT). The contract fell $1.07 to settle at $69.16 a barrel on Friday.
London Brent crude was up 12 cents at $69.04.
Pipeline bombings, attacks on oil and gas installations and kidnapping of industry workers over the past three years have prevented Nigeria from pumping much above two-thirds of its installed oil output capacity of 3 million barrels per day.
DEMAND WEAK, STOCKS HIGH
Algerian Energy and Mines Minister Chakib Khelil said on Monday oil demand was still weak due to the weakness of the U.S. and European economies and world oil stocks remained high.
Khelil told reporters ahead of a meeting of gas producers an increase in OPEC oil production was hard to envisage, despite rising crude prices.
Asian stock markets slipped on Monday with Japan's Nikkei average down 1 percent but European equities inched higher in early trade after slipping in the previous two sessions, with firmer pharmaceutical and mining stocks outpacing weaker financial shares. <.T> <.EU>
U.S. consumer confidence data on Tuesday leads a heavy calendar of economic data this week, including China's Purchasing Managers Index on Wednesday and a U.S. jobs report and manufacturing data on Thursday.
The U.S. data will help determine whether an oil market rally, which has lifted prices more than 50 percent this year on hopes of economic recovery, has any legs.
In the first big number for the week, industrial output from the world's No. 3 energy consumer Japan jumped 5.9 percent in May, the third straight month of increase after a big slump, although doubts remained whether it can sustain the momentum without a clear rebound in the United States and other key export markets.
Japanese oil demand has been hit by the economic slump over the last year. On Monday, Idemitsu Kosan Co <5019.T>, Japan's third-largest refiner, said it planned to refine 2.1 million kilolitres of crude oil in July, down 26 percent from a year earlier.
A top White House adviser said on Sunday President Barack Obama could discuss a second stimulus package to boost the economy if needed, but at the moment no more new money looked necessary.
Crude oil speculators on the New York Mercantile Exchange hiked their net long positions in the week to June 23, according to data from the Commodity Futures Trading Commission released on Friday.
(Reporting by Christopher Johnson in London and Fayen Wong in Perth; editing by Keiron Henderson)