The choice would cap an unusually public selection process, during which the president has openly discussed his views of various candidates, asked Republican senators to vote by raising their hands for those under consideration and sought the opinion of a television host. Mr. Trump also posted a video on Instagram promising “everybody will be very impressed” with his selection.

In looking past Ms. Yellen, Mr. Trump would be breaking with longstanding precedent. Every Fed chairman in modern history who completed a first four-year term was nominated for a second. The last three Fed chairmen were nominated for new terms by a president of the opposite party. And Mr. Trump has praised Ms. Yellen’s performance: During her four years, unemployment has fallen sharply, inflation has remained low and the economy is growing.

“You like to make your own mark,” Mr. Trump said last week, by way of explanation.

In choosing Mr. Powell, however, Mr. Trump would be resisting pressure by conservatives to make a larger mark on the Fed’s management of the economy. Many conservatives, including Vice President Mike Pence, favored the selection of John B. Taylor, a Stanford economist who has been an outspoken critic of the Fed’s monetary policy.

Mr. Powell, by contrast, has voted for every Fed policy decision since 2012, although he expressed some reservations in internal debates about the extent of those efforts. In recent years, he has backed the methodical unwinding of the Fed’s stimulus campaign, which involved purchasing $4 trillion worth of Treasuries and mortgage-backed securities to help the economy recover from the 2008 financial crisis.

Fed officials are often labeled hawks if they favor higher interest rates to control inflation, or doves if they want to keep rates low to promote job growth. Richard Fisher, a former president of the Federal Reserve Bank of Dallas, who worked with Mr. Powell at the Fed, said Mr. Powell “is neither a hawk nor a dove.”