WASHINGTON — The FBI’s handling of the investigation that forced CIA Director David Petraeus to resign came under new scrutiny Wednesday as FBI Director Robert Mueller faced questions on Capitol Hill and President Barack Obama alluded to lingering questions about the course of the probe.

In his first public comments on the controversy, Obama said he has seen no evidence that the scandal exposed classified information that might harm national security. But he stopped short of approving the FBI’s handling of the inquiries into the personal e-mail communications of Petraeus and U.S. Marine Corps. Gen. John Allen.

The White House and Congress were kept in the dark about the probes until election night last week. When asked at a press conference whether he should have known sooner that his CIA chief’s personal transgressions had surfaced, Obama said he was “withholding judgment with respect to how the entire process surrounding General Petraeus came up. You know, we don’t have all the information yet.”

Obama’s remarks signaled that the administration is grappling with fundamental questions surrounding an investigation that has implicated the U.S. military commander in Afghanistan, blindsided the president and still not determined whether classified material was mishandled.

Petraeus, a retired four-star general, resigned as CIA director last week after acknowledging an extramarital affair with his biographer, Paula Broadwell. Both are married.

Obama’s comments coincided with new disclosures that Broadwell had classified material and that the FBI’s initial concern centered on how an anonymous sender of menacing e-mails knew so much about the official schedules of the CIA director and the commander of allied forces in Afghanistan. The anonymous e-mails eventually were traced to Broadwell.

The messages were sent to Allen and Jill Kelley, a Florida socialite who cultivated close ties to Petraeus, Allen and other high-ranking military officers when they served at the headquarters for U.S. Central Command in Tampa, Fla. Allen is now under investigation by the Defense Department inspector general over the contents of hundreds of e-mails between him and Kelley.

The first message Allen received came in May from a sender using the alias “KelleyPatrol,” according to a person close to Kelley. The message made clear the sender knew that Allen would likely see Kelley at an upcoming event at the residence of an ambassador in Washington and that he should stay away from her, according to the person.

Subsequent messages, also anonymous, were sent to Kelley and her husband, Scott. One of them asked whether Scott Kelley knew that his wife would be meeting Petraeus in Washington at an event scheduled for the next week, the person said.

“Clearly the person knew the comings and goings of General Allen and CIA Director Petraeus,” the person said. “There was concern that someone was stalking them electronically or physically and knew the comings and goings of fairly important people.”

In mid-June, Kelley called an FBI agent she had met and told him about the e-mails. He took copies to the bureau’s Tampa office because the material showed that the sender had detailed knowledge of the travel schedules of Petraeus and Allen and because Kelley expressed concern for her safety.

The agent was identified Wednesday as Frederick Humphries, 47, who knew Kelley from a visit to her house on an unrelated case years earlier, according to law enforcement officials. Humphries was not assigned to the harassment case, but he later became frustrated over what he thought was a lagging investigation into a possible national security breach.

In late October, Humphries raised his concerns with House Majority Leader Eric Cantor, R-Va., and Cantor’s chief of staff telephoned the FBI director’s chief of staff. After Cantor’s call, the Justice Department disclosed the existence of the investigation into Petraeus to James Clapper, the director of national intelligence on Nov. 6. Clapper told the White House the next day, and Petraeus resigned last Friday.

The FBI’s last interview with Petraeus occurred on Oct. 31, and Justice Department officials have argued that they could not disclose the existence of the investigation until it was concluded.

While the law enforcement officials said the FBI did not initially believe the case raised national security concerns, officials and others told The Washington Post that Broadwell had access to the schedules of high-level officials and other information that was stamped “secret.”

A former colleague of Broadwell’s described receiving a computer disc from her several months ago that contained material marked “secret” and included both personal schedules and power-point presentations.

Broadwell’s possession and handling of such information is at the center of the FBI probe. U.S. law enforcement officials said they found a “significant amount” of classified files on Broadwell’s personal computer.