If Mr. Trump is re-elected, he would have an opportunity to replace Mr. Powell.

The Fed is led by seven governors in Washington and 12 regional presidents, who sit at central bank branches across the country. The chair and governors all hold constant and equal votes on monetary policy, but the chair has special powers. Those include marshaling the rate-setting Federal Open Market Committee, speaking directly to the public after rate-setting meetings and appearing before Congress.

Mr. Bullard dissented in June in favor of lower interest rates, making him the first voting member of the policy-setting body to do so since Mr. Powell became its leader in early 2018.

On Friday, Mr. Bullard said he would prefer a 0.25 percentage-point cut at the Fed’s next meeting, on July 30 and 31, although he prefers not to prejudge what the rate-setting committee will decide. He also indicated that the Fed might take time to assess the impact of any cut before moving again.

“Even if we make the move here, and then we’re checking the data during the fall, I think we also have to recognize that inflation is a pretty sluggish variable and it might take some time to see some impact,” he said. “I think we assess the data all the way through, up to December, and we could make judgments along the way, but I just wanted to inject some caution that it’s not just in the next 30 days.”

Market participants expect additional easing, and some investors expect a 0.50 percentage-point move as early as this month.