The person he chooses to lead the central bank will wield wide power over monetary policy and the nation's financial system, with the capacity for market-moving decisions.
Despite its ability to profoundly impact the US economy, the position operates independently of the White House. The person Trump chooses will not be beholden to the President's own views.
That makes Trump's upcoming choice a big one, particularly for a President intent on returning economic security to the Rust Belt and Midwestern states where he enjoyed electoral victory.
Trump, himself, has yet to begin serious consideration of candidates for the role, according to two people with knowledge of his thinking, though he expects to make a decision before next year. Members of Trump's team are working on preparing a list of names to present to the President soon, though they have not yet completed their internal vetting.
The current chairman of the Federal Reserve, Janet Yellen, is completing her first term, which concludes at the end of January. Whoever Trump nominates will require confirmation by the Senate. Past presidents have made their Fed chairman nominations in mid- to late October before a term expiring February 1 to allow the Senate time for hearings and votes.
The man who once led the search, National Economic Council head Gary Cohn
, was once considered virtually certain to get the job himself. Trump told an interviewer in late July that Cohn was in the running. But his fortunes have faded after he spoke out publicly against Trump's response to white nationalist violence in Charlottesville, Virginia, over the summer, and he now is no longer being considered, according to people familiar with Trump's thinking.
Trump's most recent predecessors have all renominated the sitting Fed chairman during their first terms, and Trump has not ruled out reappointing Yellen to the post, despite campaign-trail criticisms of her tenure.
"I have a lot of respect for her, and I like her," Trump told the Wall Street Journal in July
. "I would say yes, she is in the running to stay."
Trump met with Yellen in the Oval Office shortly after his January inauguration and told her she was doing a good job, according to a person familiar with their conversation. That was a sharp break from his comments about her during his run for president, when he accused Yellen of being a political hack who was attempting to bolster President Barack Obama.
He told CNBC one year ago that Yellen should be "ashamed" for keeping interest rates low while Obama was in office, which he said created a "false stock market."
This year, however, Trump has said he'd like interest rates to remain low and has trumpeted record highs on Wall Street.
Yellen, who meets regularly with Treasury Secretary Steve Mnuchin, has largely remained quiet about the first seven months of Trump's presidency. She did, however, speak out against attempts to strip financial regulations put in place following the economic meltdown, an effort that Trump has supported.
"Already, for some, memories of this experience may be fading -- memories of just how costly the financial crisis was," Yellen said at a meeting of central bankers in late August. Her remarks were seen as a rebuke of Trump's anti-regulatory agenda.
Despite those comments, people close to Trump say he has not ruled out reappointing Yellen as Fed chief. If he does rename Yellen to the post, he'd be continuing a presidential tradition of keeping a Fed chairman initially nominated by a president of the opposite party. Obama reappointed Ben Bernanke -- initially nominated by George W. Bush -- in his first year as president. Bill Clinton kept on Alan Greenspan, who was first nominated by Ronald Reagan.
Trump's aides are also preparing a list of other names should the President decide to select someone new. Two people familiar with the process say three names currently being considered are Kevin Warsh, a former Federal Reserve governor who served on the body during the 2008 financial crisis; Glenn Hubbard, the dean of the Columbia Business School who has been floated as a potential Republican Fed nominee for years; and John Taylor, an economics professor at Stanford who served in both Bush administrations.
Aside from chairman, Trump has other vacancies to fill at the Fed, including several governor seats and a recently opened vice chairman post. The current occupant of the job, Stanley Fischer, announced five days ago he was resigning next month for personal reasons.