That tradition has helped keep the Fed, with its vast power over the economy and financial system, insulated a bit from politics, but President Trump has no obligation to follow it.

The case for renominating Ms. Yellen is straightforward.

She has presided over four years of steady economic expansion and rising financial markets. She moved cautiously toward raising interest rates even though the economy seemed to be approaching full employment. By contrast, some more conservative contenders for the job have indicated they want to raise rates more quickly, which could endanger the economy as President Trump approaches midterm elections in 2018 and a potential re-election battle in 2020.

After criticizing her during his campaign, President Trump has had some kind words for her, saying in July that he liked her and that “she’s done a good job.”

Moreover, as President Trump dabbles in making deals with Democrats, reappointing Ms. Yellen could serve as an expression of good faith to Democratic senators. As administration officials focus on tax legislation and other priorities on Capitol Hill, it might be helpful to them to nominate someone who might sail through confirmation, rather than demand a bruising, time-consuming battle.

“She’s a known quantity, and there is less risk of disrupting the financial markets if the president renominates Dr. Yellen,” said Brian Gardner, managing director of Washington Research at Keefe, Bruyette & Woods, and one of the panelists. “Although President Trump is unorthodox in many ways, every president wants calm financial markets.”

The case against Ms. Yellen is similarly straightforward: She is a liberal economist in a government dominated by conservatives. She is a cerebral academic serving during the presidency of a bombastic businessman. And she is a staunch defender of the work the Fed and other bank regulators have done to try to limit risk in the financial system — including in a high-profile speech last month — amid an administration focused on deregulation.

Kevin Warsh: well connected, but with baggage

Our experts’ consensus was that Mr. Warsh has slightly better odds than Ms. Yellen. He was a Fed governor from 2006 to 2011, and was a member of Mr. Bernanke’s inner circle in shaping the Fed’s extraordinary (and often extraordinarily unpopular) actions to battle the global financial crisis.