Fears that Europe could run low on natural gas this winter eased, sending fuel prices lower, after Russian President Vladimir Putin told Gazprom PJSC to ship more gas westward.

European gas futures slid by almost 10% Thursday after Mr. Putin said the state energy giant should pump more gas to Germany and Austria as soon as it has filled stores at home. Gazprom’s underground storage sites in Western and Central Europe are almost empty at a time of year when they would normally be brimming.

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Fears that Europe could run low on natural gas this winter eased, sending fuel prices lower, after Russian President Vladimir Putin told Gazprom PJSC to ship more gas westward.

European gas futures slid by almost 10% Thursday after Mr. Putin said the state energy giant should pump more gas to Germany and Austria as soon as it has filled stores at home. Gazprom’s underground storage sites in Western and Central Europe are almost empty at a time of year when they would normally be brimming.

The Kremlin late Wednesday released a transcript of a videoconference between Mr. Putin and Gazprom Chief Executive Alexei Miller. Mr. Putin told the executive to “start consistent and planned work on increasing the amount of gas in your underground depots in Europe” as soon as Nov. 8.

“This is what we will do,” Mr. Miller replied. “This will make gas supplies more reliable and stable in the autumn and winter.”

Dutch gas futures fell 8.9% to €79.10 ($92.35) a megawatt-hour on Thursday. The move put prices on track to fall by almost a fifth in October, an unseasonal decline and the biggest monthly drop since February. U.K. futures lost more than 11%. U.S. natural-gas prices fell, too, though less than the two European benchmarks.

That Mr. Putin’s comments jolted prices shows the leverage Russia wields over Europe’s energy market, where gas is running low. Russia is the continent’s biggest supplier, and Asia has been pulling in cargoes of chilled U.S. gas that might otherwise have headed to Europe.

European gas prices are more than five times as high as they were a year ago even after Thursday’s slide, hurting consumers and holding back economies’ post-pandemic recoveries. The continent’s stores of gas are 15% below the average for the time of year, said James Huckstepp, head of European gas analysis at S&P Global Platts.

European and U.S. officials say the Kremlin is trying to pressure Europe into approving a controversial pipeline—the Nord Stream 2, which will connect Russia to Germany and bypass Ukraine—by holding back gas. Moscow has denied the accusations, and some analysts say Russia is simply behind on injecting gas into storage sites at home before its own winter chill arrives.

In 2012, the Netherlands experienced a 3.6 magnitude earthquake. It was caused by one of the world’s largest gas fields, known as Groningen, and it set off a chain of events that’s contributing to today’s sky-high energy prices. WSJ’s Shelby Holliday explains. Illustration: Sebastian Vega The Wall Street Journal Interactive Edition

It is too soon to say whether Mr. Putin’s instruction to Gazprom marks a beginning to the end of Europe’s energy crisis. The Russian president has made similar comments before, most recently on Oct. 6, but analysts say flows of gas to Europe haven’t picked up since then.

“For now the bull run seems to have stalled,” said Benedict De Meulemeester, chief executive of E&C Consultants, which advises firms on how to procure energy. “It’s a relief. The question is: How far can we trust this?”

In a sign high prices are hurting European industry, Mr. De Meulemeester said clients in energy-hungry industries have asked whether they should cut production to lower their gas and power costs.

Write to Joe Wallace at Joe.Wallace@wsj.com