The U.S. intelligence community is bracing for the possibility that former Secretary of State Hillary Rodham Clinton’s private email account contains hundreds of revelations of classified information from spy agencies and is taking steps to contain any damage to national security, according to documents and interviews Thursday.

The top lawmakers on the House and Senate intelligence committee have been notified in recent days that the extent of classified information on Mrs. Clinton’s private email server was likely far more extensive than the four emails publicly acknowledged last week as containing some sensitive spy agency secrets.

A U.S. official directly familiar with the notification, who spoke only on the condition of anonymity, said the notification of possibly hundreds of additional emails with classified secrets came from the State Department Freedom of Information Act office to the Office of Inspector General for the Director of National Intelligence.

The inspector general, the chief oversight watchdog for the entire U.S. intelligence community, subsequently sent a letter to the Republican chairmen and ranking Democrats of the Senate and House intelligence committees, the official said.

“We were informed by State FOIA officials that there are potentially hundreds of classified emails within the 30,000 provided for former Secretary Clinton,” DNI Inspector General I. Charles McCullough III late last week wrote Sen. Richard Burr, North Carolina Republican; Sen. Dianne Feinstein, California Democrat; Rep. Devin Nunes, California Republican; and Rep. Adam B. Schiff, California Democrat.

“We note that none of the emails we reviewed had classification or dissemination markings but some included IC-derived classified information and should have been handled as classified, appropriately marked and transmitted via a secure server,” Mr. McCullough wrote the four lawmakers.

The U.S. official said the intelligence community has been informed that secret information had been contained in some of Mrs. Clinton’s private emails that originated from the FBI, the DNI and the CIA as well as a spy satellite agency. It is believed the 30,000 emails remain on a thumb drive in the possession of Mrs. Clinton’s private attorney, David Kendall.

The official said the intelligence community’s first response was to take steps to secure the handling of remaining 30,000 emails and make sure they were handled on top-secret servers to avoid any further breaches, and then to assess any damage to national security from the insecure handling and release of information already in some of the publicly disseminated emails.

“Containment first, then a damage assessment is how this must be handled,” the official said.

The official said the intelligence community was already concerned, for instance, that some classified information was inadvertently disclosed by the State Department in recent weeks when one of Mrs. Clinton’s emails about Libya was publicly released.

The inspector general’s notification to Capitol Hill and the Justice Department also opens possible legal exposure for Mrs. Clinton about improper handling of classified materials, something her attorney knows much about.

Mr. Kendall represented former CIA Director David H. Petraeus last year when he pleaded guilty to a misdemeanor charge of mishandling national secrets because he gave some classified information to his mistress and biographer and stored a classified book of information in his home in an insecure manner.

Separately, Senate Judiciary Committee Chairman Chuck Grassley, Iowa Republican, sent a letter to FBI Director James Comey asking him to explain what the bureau was doing to keep secure the classified information within 30,000 Clinton emails known to be on Mr. Kendall’s thumb drive.

“It’s a serious breach of national security if the United States government fails to secure classified material in the hands of people not authorized to possess it, no matter who they are. There are fundamental questions as to what the FBI is doing to securing these classified emails and why the State Department is not fully cooperating with the inspectors general at the State Department and the Intelligence Community to ensure that all of the appropriate emails are identified,” Mr. Grassley wrote.

Mr. Grassley also sent a letter to Secretary of State John F. Kerry inquiring about the delay in sending the 30,000 emails to intelligence community inspectors general.

Mrs. Clinton, the front-runner for the Democratic presidential nomination, fended off the new questions about the email scandal and suspicious foreign donations to the Clinton Foundation, distracting from her effort to wrangle support from union bosses at the AFL-CIO’s annual summer meeting.

The former secretary of state’s email woes deepened when a federal judge scolded the State Department for delays in releasing the documents, as the agency revealed that Mrs. Clinton’s closest aides and top officials during her tenure at the agency — Huma Abedin, Cheryl Mills and Jake Sullivan — also used private email accounts and all of their message have not been turned over to the State Department.

“I think we have been proceeding in a timely fashion, and indeed the vast majority of the emails that I turned over and that are being turned over by others were already in the State Department system,” Mrs. Clinton said at a press conference at the union meeting in the Washington suburb of Silver Spring, Maryland.

Her response to reporters was the same explanation she gave in March, when it came to light that she had used a private email account exclusively for official business as America’s top diplomat, shielding her correspondence from probes by Congress and requests under the Freedom of Information Act.

It remains unclear how much of her email was captured by the State Department system during exchanges with other agency employees, especially since other high-ranking officials at the agency also were using private email accounts.

Mrs. Clinton batted questions about the email back at the agency.

“This is really a question for the State Department,” she said at the union press conference. “They are the ones that are bearing the responsibility to sort through these thousands and thousands of emails and determine at what pace they can be released, and I really hope that it will be as quickly as possible.”

Mrs. Clinton has insisted that she followed the rules and used a private email account because it was more convenient for her than juggling two smartphones. But nearly two years after she left office and after a congressional probe learned about her private email account, she turned over about 30,000 messages to the State Department and erased another 32,000 messages that she deemed personal.

At some point, she wiped clean the email server kept in her home in Chappaqua, New York, preventing any of the messages from being recovered.

Questions about her email setup and foreign donations to the Clinton Foundation while she was secretary of state, which potentially posed conflicts of interest, have dogged Mrs. Clinton’s presidential campaign. The controversies have hit her in the polls, with a majority of voters nationwide saying they don’t think she is honest and trustworthy.

The former first lady, senator and top diplomat also had to tamp down reports about increased donations to the Clinton Foundation from Swiss bank USB after she intervened to settle IRS charges that the bank had helped thousands of Americans use secret accounts to avoid U.S. taxes.

Coinciding with Mrs. Clinton’s involvement, the bank’s donations to the Clinton Foundation grew from less than $60,000 in 2008 to roughly $600,000 by the end of 2014. The bank also paid Mrs. Clinton’s husband, former President Bill Clinton, $1.5 million for participating in a series of corporate events, The Wall Street Journal reported.

In June, The Washington Times reported that Mr. Clinton’s foundation set up a fundraising arm in Sweden that collected $26 million in donations at the same time that country was lobbying Mrs. Clinton’s State Department to forgo sanctions that threatened its thriving business with Iran.

The Swedish entity, called the William J. Clinton Foundation Insamlingsstiftelse, was never disclosed to or cleared by State Department ethics officials, even though one of its largest sources of donations was a Swedish government-sanctioned lottery.

As the money flowed to the foundation from Sweden, Mrs. Clinton’s team in Washington declined to blacklist any Swedish firms despite warnings from career officials at the U.S. Embassy in Stockholm that Sweden was expanding its economic ties with Iran and potentially undercutting Western efforts to end Tehran’s rogue nuclear program, diplomatic cables show.

Mrs. Clinton said any implication of wrongdoing was “categorically false.”

“I worked hard as our nation’s first diplomat to solve problems, to work with my colleagues in government,” she said. “I remember the governmentwide efforts to try to pursue America’s interest with respect to Swiss banks and there was a resolution to that, and it continued to be the subject of diplomacy and law enforcement interest.”

She dismissed the report as routine campaign politics.

“You know, this is just the kind of unfortunate claim or charge that you see in campaigns,” Mrs. Clinton said.

The press conference demonstrated that her answers have not settled the matters and the scandals will continue to overshadow her campaign.

At the AFL-CIO meeting, Mrs. Clinton met with the union leaders behind closed doors to woo support with her pledge to fight for higher wages and for laws that would make it easier to unionize workplaces.

“I asked for their support going forward. I asked them to be my partner in making sure that we stand against those powerful forces on the other side that don’t agree with the [union] agenda,” she said.

Union insiders say the leadership prefers more aggressively pro-union candidates such as Sen. Bernard Sanders, the Vermont independent and avowed socialist who has emerged as the chief rival to Mrs. Clinton.

But Mr. Sanders trails by a wide margin and is widely viewed as unable to win.

That leaves the unions stuck with Mrs. Clinton, the all-but-inevitable nominee who has refused to take a position on the Keytsone XL oil pipeline or the pending trade deal with Pacific Rim countries — issues that are top priorities for unions.

“That’s what we’re waiting for — for her to take a stand,” said an official from a union local at the meeting.

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