The Fed, which raised rates for five consecutive quarters amid a roaring economy with the lowest unemployment rate in nearly two decades, has since paused its campaign. Mr. Powell has repeatedly rebuffed any suggestion that the central bank is being influenced by Mr. Trump’s broadsides, saying it is adopting a more “patient” approach to interest rates given signs of economic weakness in the United States and abroad.

With two open seats on the seven-member Fed, Mr. Trump has the ability to drastically shape the central bank for the foreseeable future. The president has already appointed four of the Fed’s current governors, who carry 14-year terms. One governor, Lael Brainard, is a holdover from the Obama administration.

The appointment and confirmation of Mr. Moore and Mr. Cain could put the Fed in uncharted territory. While the institution has strongly rooted values around technical competence and apolitical debate, Mr. Trump’s latest choices have been political actors rather than in-the-weeds experts in any of the main areas in which the Fed makes policy.

Two governors alone cannot entirely shape Fed policy, but they can have an effect on decisions. The Federal Open Market Committee, which sets interest rates, consists of 12 voting members, including the seven governors, the president of the Federal Reserve Bank of New York and four regional Federal Reserve Bank presidents, who serve one-year terms on a rotating basis. Changes to interest rates must be approved by a majority of the voting members.

Both Mr. Moore and Mr. Cain are undergoing background checks, and if nominated, must be confirmed by the Senate.

The choice of such candidates to fill two of the most powerful jobs in economic policy is raising fears among some Fed veterans that the president is installing political allies at the central bank to do his bidding.

“It’s one thing to put cronies into the executive branch, but I think the central bank is another kind of institution that it’s so critical in that it holds the reins solely to monetary policy,” said Sarah Bloom Raskin, who served on the Fed’s board of governors from 2010 to 2014. “It has the potential to undermine the credibility of monetary policy.”