WASHINGTON — High-level officials at the Federal Bureau of Investigation and the Justice Department were notified in the late summer that FBI agents had uncovered what appeared to be an extramarital affair involving the director of the Central Intelligence Agency, David Petraeus, government officials said Sunday.

But law enforcement officials did not notify anyone outside the FBI or the Justice Department until last week because the investigation was incomplete and initial concerns about possible security breaches, which would demand more immediate action, did not appear to be justified, the officials said.

The new accounts of the events that led to Petraeus’ sudden resignation on Friday shed light on the competing pressures facing FBI agents who recognized the high stakes of any investigation involving the CIA director but who were wary of exposing a private affair with no criminal or security implications. For the first time Sunday, the woman whose report of harassing emails led to the exposure of the affair was identified as Jill Kelley, 37, of Tampa, Fla.

Some members of Congress have protested the delay in being notified of the FBI’s investigation of Petraeus until just after the presidential election.

Sen. Dianne Feinstein, D-Calif., the chairwoman of the Intelligence Committee, said Sunday that her committee would “absolutely” demand an explanation. An FBI case involving the CIA director “could have had an effect on national security,” she said on Fox News Sunday. “I think we should have been told.”

A close friend of the Petraeus family said Sunday that the intimate relationship between Petraeus and his biographer, Paula Broadwell, began after he retired from the military last year and about two months after he began work as CIA director. It ended about four months ago, said the friend, who did not want to be identified while discussing personal matters. In a letter to the CIA workforce on Friday, Petraeus acknowledged having the affair. Broadwell has not responded to repeated requests for comment.

Under military regulations, adultery can be a crime. At the CIA, it can be a security issue, since it can make an intelligence officer vulnerable to blackmail, but it is not a crime.

The same Petraeus family friend confirmed on Sunday the identity of Kelley, whose complaint to the FBI about “harassing” emails, eventually traced to Broadwell, set the initial investigation in motion several months ago. Kelley, along with her husband, became friends with Petraeus and his wife, Holly, when Petraeus was head of the military’s Central Command, which has its headquarters at MacDill Air Force Base in Tampa.

“We and our family have been friends with General Petraeus and his family for over five years,” Kelley and her husband, Scott Kelley, said in a statement released Sunday. “We respect his and his family’s privacy, and want the same for us and our three children.”

The involvement of the FBI, according to government officials, began when Kelley, alarmed by about half a dozen anonymous emails accusing her of inappropriate flirtatious behavior with Petraeus, complained to an FBI agent who is also a personal friend.

Because the sender’s account had been registered anonymously, investigators had to use forensic techniques to identify who was writing the emails.

Eventually they identified Broadwell as a prime suspect and obtained access to her regular email account. In its inbox, they discovered intimate and sexually explicit emails from another account that also was not immediately identifiable. Investigators eventually ascertained that it belonged to Petraeus and studied the possibility that someone had hacked into Petraeus’ account or was posing as him to send the explicit messages.

Eventually they determined that Petraeus had indeed sent the messages to Broadwell and concluded that the two had had an affair. Then they turned their scrutiny on him, examining whether he knew about or was involved in sending the harassing emails to Kelley.

FBI agents interviewed Broadwell for the first time the week of Oct. 21, and she acknowledged the affair, a government official briefed on the matter said. She also voluntarily gave the agency her computer. In a search of it, the agents discovered several classified documents, which raised the additional question of whether Petraeus had given them to her. She said that he had not. Agents interviewed Petraeus the following week. He also admitted to the affair but said he had not given any classified documents to her. The agents then interviewed Broadwell again on Friday, Nov. 2, the official said.

Based on that record, law enforcement officials decided there was no evidence that Petraeus had committed any crime and tentatively ruled out charges coming out of the investigation, the official said. Because the facts had now been settled, the agency notified James Clapper, the director of national intelligence, about 5 p.m. on the following Tuesday — Election Day.

Meanwhile, the FBI agent who had helped get a preliminary inquiry started and learned of Petraeus’ affair and initial concerns about security breaches became frustrated. Apparently unaware that those concerns were largely resolved, the agent alerted the office of Rep. Eric Cantor, the House majority leader, about the inquiry in late October. Cantor passed on the concerns to Mueller.

Cantor revealed Saturday that he had talked with the FBI agent.

Officials said Sunday that the timing of notifications had nothing to do with the election, noting that there was no obvious political advantage for either President Barack Obama or Mitt Romney in the news that the CIA director had had an affair; Petraeus is highly regarded by both Republicans and Democrats.

John Prados, a historian and author on intelligence and its abuses, said the case “posed several dilemmas for the FBI” that would have prompted agents and their bosses to proceed gingerly.

“Petraeus is a very important person, so they would want to be crystal-clear on exactly what happened and what the implications were,” Prados said.

But if the security issues were resolved and no crime had been committed, Prados said, there was no justification for informing Congress or other agencies that Petraeus had an affair.

“In my view, it should never have been briefed outside the bureau,” he said.