LONDON—Oil prices fell on Friday, in response to concerns over resilient U.S. crude production.

The number of U.S. oil-drilling rigs rose by 12 to 640 in the past week, snapping 29 straight weeks of decline, data from Baker Hughes BHI 0.57 % showed late on Thursday. The rig-count, which some investors see as a proxy for activity in the oil industry, has fallen sharply since oil prices headed south last year. U.S. oil output, however, has remained strong and continued to pressure oil prices.

Brent crude, the global oil benchmark, fell 0.7% to $61.62 a barrel on London’s ICE Futures exchange. In a shortened trading session because of the U.S. Independence Day holiday, U.S. crude futures were trading down 0.7% at $56.55 a barrel on the New York Mercantile Exchange.

“The pickup in activity might be illustrative of the competitiveness of the U.S. shale industry which, thanks to cutting costs, might have become comfortable with producing oil at prices around $60 per barrel,” said Norbert Ruecker, head of commodities research at Julius Baer. JBAXY 0.53 %

“The U.S. shale industry has become the oil market’s swing producer and its responsiveness to prices will shape the market for the years to come, paving the way for lower prices for longer,” he said.

Though this is the first increase since 2014, there are still about 60% fewer rigs working since a peak of 1,609 in October.

“One swallow doesn’t make a summer,” said analysts at Commerzbank. CRZBY -0.38 % But the bank noted if other new oil rigs were to be added in the coming weeks, that would cast doubt over the expected decline in U.S. shale oil production.

“Yet precisely such a decrease is necessary so that the oversupply on the global oil market can be reduced and oil prices can further recover,” Commerzbank said.

According to analysts at Deutsche Bank, DB 1.19 % U.S. shale oil production will decline over coming months, likely into early 2016. However, it said that the decline would prove relatively shallow, owing to the continued benefits of productivity gains.

That would leave the oil market materially oversupplied in the first half of next year, by an average of 2.3 million barrels a day. Currently, the market is oversupplied by between 1.5 million and 2 million barrels a day, according to analysts’ estimates.

Meanwhile, the Iran nuclear talks continue to drag on, but Western officials are eager to reach a deal by Tuesday, after a one-week extension to the previous deadline.

A final agreement to curb Iran’s nuclear program is expected to pave the way for the lifting of Western sanctions and release more Iranian crude on the oversupplied global market.

“Although oil is unlikely to flow at substantial volumes before 2016, Iran remains a long-term bearish factor for the market,” Mr. Ruecker said.

Investors are also tracking the latest twists and turns in the Greek debt crisis. This weekend’s referendum on the conditions required by the country’s creditors for more funding, is likely to dominate market sentiment next week.

Nymex reformulated gasoline blendstock for August—the benchmark gasoline contract—fell 0.4% to $2.0265 a gallon, while ICE gas oil for July changed hands at $564.50 a metric ton, down $9.75 from Thursday’s settlement.

Write to Georgi Kantchev at georgi.kantchev@wsj.com

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