WASHINGTON—The Federal Reserve indicated that the economy has made progress toward the central bank’s employment and inflation goals, and officials offered a hint they could begin to reduce their asset purchases later this year.

The Fed since the end of last year has said its monthly purchases of $120 billion in bonds would continue until the economy achieves “substantial further progress” toward the Fed’s goals of low unemployment and inflation reaching 2%. On Wednesday, the Fed said that “since then, the economy has made progress toward these goals, and the Committee will continue to assess progress in coming meetings.”

Officials had been set to deepen their deliberations over how and when to begin paring, or tapering, their asset purchases, which they initiated in March 2020 to quell an incipient market panic. With their short-term benchmark interest rate pinned near zero, they have continued to purchase Treasurys and mortgage-backed securities at the current pace since June 2020 to provide additional stimulus by holding down longer-term interest rates.

The Fed’s statement also modestly upgraded its assessment of the economy. “With progress on vaccinations and strong policy support, indicators of economic activity and employment have continued to strengthen,” it said. “The sectors most adversely affected by the pandemic have shown improvement but have not fully recovered,” it said. Officials in June had described those sectors as remaining weak.

Supply-chain bottlenecks have driven inflation to higher levels than many economists expected this year, and those readings have raised “the possibility that inflation could turn out to be higher and more persistent than we expect,” said Fed Chairman Jerome Powell at a news conference after Wednesday’s meeting.

Some officials are concerned that a burst of inflation this year from bottlenecks associated with reopening the economy will prove more durable than previously anticipated. These policy makers are eager to start the taper, in part because they and their colleagues have said they aren’t likely to consider raising interest rates until they are done tapering the asset purchases.

Another camp thinks recent price pressures will subside and could leave the Fed in the same position that it faced for much of the past decade, in which global forces kept inflation below 2% even with historically low interest rates. They are worried that accelerating plans to wind down the asset purchases could raise questions among investors about the Fed’s commitment to achieving its economic goals.

Mr. Powell has pledged to provide ample notice to financial markets before the Fed starts tapering to avoid catching investors by surprise. In April, he said the Fed was “a long way from” its tapering goals, and he characterized the economy as “still a ways off” from them in June.

Officials also must consider the pace of any reductions. During a prior asset-purchase program that ended in 2014, the Fed shrank its purchases in modest, equal amounts over the course of 10 months. It then waited another 14 months before raising interest rates.

Another tactical question centers on whether to reduce the pace of purchases of Treasurys and mortgage-backed securities equally. Some officials have raised concerns about rising home prices and are pressing to stop purchases of mortgage bonds sooner.

But Mr. Powell and other officials have poured cold water on those concerns in recent weeks. They have said mortgage buying, by purchasing longer-dated assets, provides a way to more broadly stimulate the economy and isn’t focused squarely on housing markets.

For a third straight month in June, inflation ran hotter than many economists had expected. The Labor Department’s consumer-price index increased 5.4% from a year ago, the highest 12-month rate since August 2008.

Mr. Powell said two weeks ago that many of the elevated price pressures can still be traced to goods and services affected by supply-chain bottlenecks and other pandemic-driven upheaval. As a result, he said it would be too soon for the Fed to abandon its earlier expectation that prices will return to their 2% target on their own and to raise rates to cool down demand and reduce inflation faster.

But investors have questioned how long the central bank and its 12-member rate-setting committee would need to revisit those assumptions if price pressures don’t abate soon. Price pressures in some sectors of the economy where inflation had been subdued over the past year, including residential rents, have picked in recent months.

Since Fed officials last met in June, government-bond prices have jumped, a sign that investors are less confident about long-term growth prospects and less worried about inflation.

Yields, which rise when bond prices fall, climbed sharply earlier in the year, lifted by expectations that vaccinations and fiscal stimulus would spur an economic boom. After hitting a 13-month high of 1.75% at the end of March, the 10-year Treasury yield has declined—to 1.57% on June 16, after the Fed concluded its previous meeting, and to 1.26%, near a five-month low, at noon on Wednesday.

Write to Nick Timiraos at nick.timiraos@wsj.com