WASHINGTON — President Trump delights in making spectacles out of personnel decisions. He conducted cabinet interviews at his New Jersey golf club, inviting members to gather and gawk. He summoned both finalists for a Supreme Court seat to the White House on the day of the announcement. And now he is conducting the most dramatic and drawn-out search for a Federal Reserve chairman in the long history of that stolid institution.

Mr. Trump is very publicly deliberating between two candidates with strikingly different views about the practice and purpose of monetary policy: Jerome H. Powell, a Fed governor who has voted in favor of every Fed policy decision since 2012, and John B. Taylor, a Stanford economist who is among the Fed’s most vocal critics.

The president also continues to insist that he could decide to renominate the Fed’s chairwoman, Janet L. Yellen, whose four-year term ends in February.