It is unclear exactly when the F.B.I. renewed its interest in the Clinton Foundation, or whether agents were instructed by anyone in Washington to start investigating again. But the F.B.I.’s decision to take additional investigative steps is sure to outrage Democrats who will see the inquiry as an attempt by Attorney General Jeff Sessions to placate the president.

As Mr. Mueller’s investigation has intensified, the president and his conservative allies have mounted blistering counterattacks trying to discredit the F.B.I. and federal prosecutors. Mr. Trump has described the investigation as a witch hunt and accused F.B.I. leadership under the bureau’s former director, James B. Comey, of being biased toward Mrs. Clinton.

Some congressional Republicans have sought to cast doubt on an explosive dossier of unsubstantiated claims about Mr. Trump. On Friday, two influential Republican senators asked the Justice Department to investigate whether the author of the dossier, Christopher Steele, a respected former British spy, lied to the federal authorities.

Mr. Trump’s calls to investigate Mrs. Clinton break with longstanding presidential practice. Since the Watergate scandal, the Justice Department has conducted criminal investigations largely free of political influence from the White House. Mr. Trump, by contrast, has declared that he has an “absolute right to do what I want to do with the Justice Department.”

The Clinton Foundation dismissed the investigation as politicized.

“Time after time, the Clinton Foundation has been subjected to politically motivated allegations, and time after time, these allegations have been proven false,” Craig Minassian, a spokesman for the foundation, said in a statement.