Clinton aides did not respond to a request for comment about Mr. Trump’s remarks.

During their second debate, Mr. Trump turned to Mrs. Clinton and vowed, “If I win, I am going to instruct my attorney general to get a special prosecutor to look into your situation, because there’s never been so many lies, so much deception.”

After Mr. Trump’s turnaround on Tuesday, the conservative commentator Ann Coulter, one of his staunchest supporters during the campaign, suggested on Twitter that Mr. Trump was overstepping his role.

“Whoa! I thought we elected @realDonaldTrump president,” she wrote. “Did we make him the FBI, & DOJ? His job is to pick those guys, not do their jobs.”

Tom Fitton, president of Judicial Watch, a conservative advocacy group that has been a relentless critic of Mrs. Clinton, said on Tuesday that it would be a mistake for Mr. Trump to drop the threat of appointing a special counsel to look into her email use.

Mr. Fitton said that although the F.B.I. had already investigated the matter twice, that inquiry was flawed. His group has gained access to thousands of pages of Mrs. Clinton’s State Department emails through lawsuits and is pressing its case in court even after her defeat.

Representative Jason Chaffetz, a Utah Republican who leads the House Oversight and Government Reform Committee, said last week on Fox News that his committee would continue to investigate the matter as well because “we’ve got to get to the truth.” Mr. Chaffetz’s office said on Tuesday he was unavailable for comment about Mr. Trump’s statements.

Although the F.B.I.’s email investigation is closed, the agency still has an open inquiry into the Clinton Foundation. That inquiry began after the 2015 publication of “Clinton Cash,” a book by Peter Schweizer that asserted that some foreign entities had given money to the foundation in return for State Department favors when Mrs. Clinton was there. The Clintons have denied the assertions.