Yellen has overseen a delicate transition in U.S. monetary policy. She ended quantitative easing, raised the benchmark lending rate off zero, and has now started to unwind the central bank’s $4.5 trillion balance sheet with little impact on the economy or financial markets.

She has not presided over a recession, unlike her three predecessors. Unemployment has declined on her watch to 4.2 percent from 6.7 percent, while inflation is low at 1.4 percent. U.S. stock markets have hit record highs -- a favorite observation by the president -- and interest rates remain unusually low in an expansion now in its ninth year. Her term ends in February.

Yellen fared well during her interview with Trump and is respected by the president and other administration officials including Mnuchin. Even so, she has no constituency among Trump advisers in favor of her reappointment, and she also faces questions from some Republicans in Congress about her stewardship of the Fed.

Congressional Republicans

On Thursday, Senate Banking Committee Chairman Mike Crapo said Yellen had been too slow to raise rates, though he declined to weigh in on whether she should get another term as chair. The nomination would require confirmation by the Senate.

Representative Warren Davidson, an Ohio Republican and member of the conservative House Freedom Caucus, is circulating a letter for colleagues on the House Financial Services Committee to sign against Yellen’s re-appointment.

Warsh and Powell were interviewed at the White House last month. Some conservatives have concerns about Powell, a member of the Fed board of governors who was appointed by former President Barack Obama.

Warsh, meanwhile, lacks support from Mnuchin, according to two other people familiar with the process, though they would not say why. His tenure on the Fed board has been criticized by a diverse group of economists ranging from Scott Sumner to Nobel laureate Paul Krugman.

Taylor impressed Trump after an hour-long meeting at the White House last week, several people familiar with the matter said.

Trump has always been partial to hiring people with whom he has a good relationship. However, Trump told the Wall Street Journal in July that he would “like to see rates stay low,” and Taylor is the namesake of a well-known monetary policy rule that would generally call for higher interest rates.