Uncertainty about Mr. Trump’s decision is already causing some unease in financial markets. Ms. Yellen has charted a steady course for monetary policy in the coming years: gradual increases in interest rates and a gradual reduction in the Fed’s bond holdings. But some of the candidates Mr. Trump is considering, including Mr. Warsh and Mr. Taylor, have argued that the Fed should raise rates and reverse its postcrisis buildup of bonds more quickly.

The Fed chairman also plays a leading role in financial regulation and will have to navigate calls by Republican leaders, including Mr. Trump, to roll back some of the rules put in place after the 2008 financial crisis. Ms. Yellen was a principal architect of those more stringent regulations and has said the financial system is much stronger as a result. Mr. Trump has criticized many of the banking rules, saying they are excessive and impeding economic growth. The candidates to replace Ms. Yellen all share those concerns to varying degrees.

Given the high stakes, Mr. Trump’s decision has become a source of intrigue in political and economic circles this year. There is also a growing sense of urgency: Both Ms. Yellen and her predecessor, Ben S. Bernanke, were nominated in October, leaving ample time for the Senate to hold confirmation hearings. Mr. Trump has said he does not plan to make a decision until later in the year, leaving little time for the Senate to act given the tight calendar. Aides say he is not yet focused on the search. Additional candidates could ultimately be considered.

The administration officials and the other person with knowledge of the search requested anonymity because of the sensitive nature of the job in question. A White House spokeswoman had no immediate comment, and Mr. Cohn, Mr. Taylor and Mr. Warsh did not respond to requests for comment. A Fed spokeswoman said Ms. Yellen and Mr. Powell declined to comment.

The next Fed chairman is likely to begin the four-year term in an atmosphere of relative tranquillity during one of the longest economic expansions in American history. But the Fed is confronting a number of difficult issues. One question is whether it can do more to increase economic growth. Another is whether the Fed will be ready for the next economic downturn.