The F.B.I. has concluded two times, including after a surprise review that began just 11 days before the election, that Mrs. Clinton should not face criminal prosecution over her handling of her private email server. Mrs. Clinton on Saturday blamed her loss in part on the F.B.I.’s last-minute intervention.

Even so, legal analysts said there was little doubt that as president, Mr. Trump would have the power to direct his attorney general — Rudolph W. Giuliani has been frequently mentioned for the job — to appoint an outside special counsel to reinvestigate the matter in light of new evidence that may have developed.

In an interview with The Wall Street Journal that was published on Friday, Mr. Trump deflected a question about naming a special prosecutor.

“It’s not something I’ve given a lot of thought, because I want to solve health care, jobs, border control, tax reform,” he said.

But his top aides have left the door open to such a move.

In an appearance Thursday on Fox News, Mr. Giuliani — a top adviser to Mr. Trump on legal and national security issues — said he did not think that Mr. Obama should pardon Mrs. Clinton, and he raised the prospect that the Trump administration could investigate not only the email server, but also the Clinton Foundation, the family charity. Several F.B.I. offices are known to have examined questions about the charity’s acceptance of gifts from foreign leaders, but those inquiries appear to have been paused.

Mr. Giuliani said that during the campaign Mr. Trump had “talked about an independent counsel doing it, who would be not a Republican, not a Democrat, somebody free of any political question.”

The decision about appointing a special prosecutor is a “tough” one, Mr. Giuliani said in a separate appearance on CNN. “It’s been a tradition in our politics to put things behind us,” he said. “On the other hand, you have to look at, how bad was it?”