Earlier in the year, it appeared that chief White House economic advisor Gary Cohn, the former chief operating officer at Goldman Sachs, was the front-runner. However, Cohn's criticism of Trump's response to the racial violence in Charlottesville, Virginia, apparently alienated him from the president.
PredictIt, and online site where users can place bets on the probability of various events, had put Yellen as a prohibitive leader in the race for the appointment, but that changed quickly Friday. Warsh vaulted to the front of the pack, most recently carrying a 36 percent chance of getting the job.
CNBC has reached out to Warsh for comment.
In addition to serving with the Fed, Warsh also was an on advisory council to Trump that disbanded in August following the Charlottesville furor. Warsh also served as an advisor to former Florida Gov. Jeb Bush, whom Trump defeated in the Republican primaries.
Trump's relationship with the Fed has vacillated. During the campaign he bitterly criticized Yellen, saying she was using monetary policy to prop up the economy in order to make former President Barack Obama look good. But during his time in office, Trump has softened his stance toward Yellen, saying he respects her.
Warsh is expected to be more hawkish than Yellen on monetary policy and, perhaps more importantly, likely to favor a less restrictive regulatory environment for banks.
The Fed currently is in the midst of a gradual pace of rate hikes and is about to begin rolling off some of the $4.5 trillion balance sheet of bonds that it acquired in its economic stimulus efforts. On the regulatory side, the Yellen Fed has kept a tight watch on banks, particularly on risk-taking and capital levels.
Trump has said over-regulation of banks has held back lending and economic growth.