“If we got to the point where we were comfortable on the public health side that the pandemic was largely behind us, and was not going to resurge in some way that was surprising, then I think we could talk about adjusting monetary policy,” St. Louis Fed President James Bullard told reporters after a speech Wednesday. “I don’t think we’re quite to that point yet, but it does seem like we’re getting close.”

Mr. Bullard doesn’t currently vote on the Fed’s policy-setting committee, but his remarks were echoed by other officials who do.

Atlanta Fed President Raphael Bostic, a voting member of the Fed’s rate-setting committee, made similar remarks in a Bloomberg television interview.

“We’re going to have to be very nimble in terms of our monitoring of the economy and our policy responses,” Mr. Bostic said Wednesday.

The latest public remarks came ahead of the Fed’s release of minutes from its April 27-28 meeting. The minutes showed general agreement among officials on the need to continue supporting the economy with near-zero interest rates and bond purchases.

But they also dropped the Fed’s first hint that policy makers could soon begin discussing a slowdown in the pace of its Treasury and mortgage-bond purchases, which currently total at least $120 billion a month.

“A number of participants suggested that if the economy continued to make rapid progress toward the committee’s goals, it might be appropriate at some point in upcoming meetings to begin discussing a plan for adjusting the pace of asset purchases,” the minutes said. They noted that many officials echoed Chairman Jerome Powell’s view that the Fed should give markets plenty of advance warning before it begins reducing the purchases.

Long-term bond yields jumped and stocks extended losses after the minutes were released. Yields on 10-year Treasury notes rose as high as 1.692%, up from 1.62% on Wednesday morning.

“The Fed has repeatedly said it will provide a long runway of guidance before tapering begins,” said Tim Duy, chief economist at SGH Macro Advisors, referring to the scaling back of asset purchases. “This is the front end of that runway.”

The April meeting took place before economic data showed a surge of inflation, a slower-than-expected pace of hiring and mounting supply constraints. Fed officials have said for months that they think higher inflation this year will be temporary, allowing them to maintain easy-money policies until the labor market more fully recovers from the pandemic.

“However, a couple of participants commented on the risks of inflation pressures building up to unwelcome levels before they become sufficiently evident to induce a policy reaction,” minutes from the April meeting said.

The Labor Department reported last week that the consumer-price index, a closely watched measure of inflation, rose 4.2% in the 12 months through April. So-called core prices, which exclude volatile components such as food and energy, rose 0.9% in April from March, the fastest one-month gain since 1981 and significantly more than economists had expected.

Fed officials voted unanimously in April to continue buying at least $120 billion of Treasury and mortgage bonds each month and to hold overnight interest rates near zero.

The policies are intended to reduce borrowing costs for consumers and businesses to help spur economic growth and a quicker recovery in the labor market, which is 8 million jobs short of its pre-pandemic level.

The Fed said in a statement after the April meeting that its asset purchases will continue at the current pace until the economy makes “substantial further progress” toward its goals of maximum employment and 2% average inflation. Overnight interest rates won’t be raised until those objectives are fully reached, an even higher bar.

“In my judgment, through that April employment report we have not made substantial further progress,” Fed Vice Chairman Richard Clarida said on Monday. While he said he believes most of the recent rise in inflation is likely to be transitory, Mr. Clarida also said central bankers “have to be attuned and attentive to the incoming data.”

“We’re in a very fluid period,” he said.

Fed Gov. Randal Quarles, testifying Wednesday before a House panel, said that the central bank’s best analysis indicates that price pressures will be “temporary even if significant” as the economy reopens. If that proves wrong and high inflation appears more lasting, the Fed is prepared to rein it in.

“But I do think that if we were to try now to stay ahead of the inflation curve we could end up significantly constraining the recovery,” Mr. Quarles said.