* Colonial aims for substantial resumption in operations this week

* COVID-19 outbreak hits Indian fuel demand

* U.S. crude inventories fall, gasoline stocks rise: API

NEW YORK, May 11 (Reuters) - Oil prices settled higher on Tuesday, as lingering fears of gasoline shortages due to an outage at the largest U.S. fuel pipeline system after a cyber attack brought futures back from an early drop of more than 1%.

Brent crude futures rose 23 cents, or 0.3%, to settle at $68.55 a barrel while U.S. West Texas Intermediate (WTI) crude futures rose 36 cents, or 0.6%, to end the session at $65.28.

Benchmark gasoline futures prices ended the session 0.3% higher at $2.1399 a gallon.

On Monday, Colonial Pipeline, which transports more than 2.5 million barrels per day (bpd) of gasoline, diesel and jet fuel, said it was working to restore much of its operations by the end of the week.

"While the short-term risk is being played down, the market is still visibly shaken by the event, given the nature of the attack and the scale of the infrastructure," said Rystad Energy's oil markets analyst Louise Dickson.

"The market is now concerned about the likelihood of such an event being repeated and about the severity of future attacks."

Fuel supply disruption has driven gasoline pump prices to multi-year highs and demand has spiked in some areas served by the pipeline as motorists fill their tanks.

"With little information forthcoming from the private company, the market appears to be proceeding on the assumption that normal flows will resume by the upcoming weekend and since no operational problems appear to exist, this guidance would appear correct." said Jim Ritterbusch, president of Ritterbusch and Associates.

U.S. crude oil inventories fell by 2.5 million barrels in the most recent week, according to two market sources on Tuesday, citing American Petroleum Institute figures. . Gasoline inventories rose by 5.6 million barrels

Traders booked at least four tankers to store refined oil products off the U.S. Gulf Coast refining hub after a cyber attack that knocked out the pipeline, shipping data showed on Tuesday.

North Carolina, the U.S. Environmental Protection Agency and Department of Transportation issued waivers allowing fuel distributors and truck drivers to take steps to try to prevent gasoline shortages.

OPEC on Tuesday raised its forecast for demand for its crude by 200,000 bpd and stuck to its prediction of a strong recovery in global oil demand this year as growth in China and the United States counters the coronavirus crisis in India.

Meanwhile, the rapid spread of infections in India has increased calls to lock down the world's second-most populous country and the third-largest oil importer and consumer.

India's top state oil refiners have already started reducing runs and crude imports as the new coronavirus cuts fuel consumption, company officials told Reuters on Tuesday.

(Additional reporting by Devika Krishna Kumar, Julia Payne and Shadia Nasralla in London; Shu Zhang and Sonali Paul; Editing by David Gregorio Editing by Gabriela Baczynska, David Goodman and David Gregorio)