U.S. oil prices fell for a third straight session on Friday, a signal to some in the market that the recent strong rally in crude-oil prices may be petering out.

Light, sweet crude for June delivery settled down 19 cents, or 0.3%, to $59.69 a barrel on the New York Mercantile Exchange. Prices rose 0.5% in the week.

Brent crude for July delivery, the global benchmark, rose 11 cents, or 0.2%, to $66.81 a barrel on the ICE Futures Europe exchange. The contract rose 1% on the week.

Oil prices have climbed off near-six-year lows since March on expectations that spending cutbacks would lead to a drop in U.S. crude-oil production. The number of rigs drilling for oil in the U.S. has plummeted in the past few months, spurring expectations of a decline in output.

But U.S. shale-oil companies say that if prices rise to around $65 a barrel, they can add rigs and increase production.

"I keep seeing guys talking about production dropping off a shelf, just crumbling, and I don't see it happening," said Michael Hiley, an energy trader at brokerage LPS Partners Inc. "If [prices] go up, you're going to get new production. If [prices] go down, you're going to get less production. You have elasticity of supply that's bounding this market."

The number of rigs drilling for oil in the U.S. fell by eight to 660 this week, according to oil-field-services firm Baker Hughes Inc. Though that marks the lowest level since August 2010, this week's decline was the smallest of the year.

In a key shale-drilling region in Texas, the Eagle Ford, the number of rigs increased by one in the week.

Oil prices also largely shrugged off an incident involving an oil tanker in the Middle East. Iranian patrol boats on Thursday opened fire on a Singapore-flagged vessel as it moved through the Strait of Hormuz.

"There is a saying that bull markets don't end due to bad news, they end when the market stops reacting to good news," said fuel distributor TAC Energy in a note. "It's early yet to say we've seen the end of the impressive 8 week rally [in U.S. oil prices]...but the signs are beginning to appear."

Money managers including hedge funds cut their aggregate bet on rising U.S. oil prices by 2.1% in the week ended Tuesday, according to data from the Commodity Futures Trading Commission released Friday.

Gasoline futures settled down 0.07 cent at $2.0568 a gallon, but posted a 3.3% weekly gain. Diesel futures fell 0.08 cent to $2.0048 a gallon. Prices rose 2.6% on the week.

Write to Nicole Friedman at nicole.friedman@wsj.com

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