Trump harshly criticized Yellen during campaign, saying at one point "I think she is very political and to a certain extent, I think she should be ashamed of herself. Because it is not supposed to be that way,'' he said.
Few Fed observers could recall a presidential nominee so openly and harshly criticizing the Fed or the Fed chair, making some question whether Yellen can continue in the job.
"I think it's plausible that she would find this an intolerable situation where she not only doesn't have the confidence of the president, but that he thinks she's undermined the credibility and independence of the Fed," said Jimmy Pethokoukis, an economic policy analyst at the American Enterprise Institute and a CNBC contributor.
The Fed declined to comment.
Presidents meet periodically with Fed chairs and Fed chairs meet frequently with Treasury secretaries. Given what's been said by Trump, unless an olive branch is extended, such meetings would seem awkward at best and dysfunctional at worst.
How Trump handles his relationship with Yellen could ultimately determine what she decides to do. Continuing criticism would appear to make her position untenable, especially since the Federal Reserve has few allies in Congress. It will also be among the most critical bellwethers of how the president-elect navigates repairing relationships with the many targets of his harsh campaign.