The FBI went to bat for Hillary Clinton on Tuesday, saying that handling emails with classified information — even when it’s marked — isn’t enough to prove she had “knowledge or intent” to break the law.

Even as it turned over agents’ interview notes and other files from the investigation into Mrs. Clinton to House investigators, the FBI defended its conclusion that Mrs. Clinton wasn’t likely to be convicted of mishandling classified information, despite appearing to have broken the letter of the law.

Instead, Jason V. Herring, the FBI’s chief liaison to Capitol Hill, said the Justice Department has long questioned whether the law against “grossly negligent” handling of classified material would stand up in court, particularly when the person being charged swears he or she didn’t intend to break the law.

In particular, Mr. Herring explained that three messages Mrs. Clinton handled that were marked with a “(C)” indicating classified information weren’t enough proof to justify a prosecution. He said it’s unclear whether the information was secret at the time, despite the markings, and that Mrs. Clinton merely handled it and wasn’t the original sender.

“The fact that Secretary Clinton received emails containing ‘(C)’ portion markings is not clear evidence of knowledge or intent,” Mr. Herring said in a letter to House Oversight and Government Reform Committee Chairman Jason Chaffetz, Utah Republican, and Rep. Elijah E. Cummings of Maryland, the panel’s ranking Democrat.

A year and a half after her secret emails became public, Mrs. Clinton is fending off questions about her decision to forgo an official State Department account and instead use an account tied to a server she kept at her home in New York, which authorities say lacked competent security.

The FBI early last month recommended against prosecuting her for violating secrecy laws. Agency Director James B. Comey said that while Mrs. Clinton was “negligent,” he couldn’t prove “gross negligence.” He said he concluded that Mrs. Clinton wasn’t “sophisticated” enough to understand the “(C)” markings or the risks she was running with national security by keeping top-secret information stored at her home.

Still, top congressional Republicans say Mrs. Clinton could be prosecuted for perjury, accusing her of lying to Congress in her own 2015 testimony.

Mr. Chaffetz and House Judiciary Committee Chairman Bob Goodlatte, Virginia Republican, laid out that case in a letter Monday to the Justice Department, pointing to four areas where they said Mrs. Clinton misled: her claim that she had turned over all of her work-related messages; her statement about the number of servers or devices she used to handle email; her declaration that her attorneys read every single message to determine which were work-related; and her insistence that none of her messages was marked classified.

The congressmen said the “(C)” markings disproved that final claim by Mrs. Clinton.

Of the three messages with those markings, one has been to contain information that is classified at this time. The State Department has not said whether it should have been classified at the time Mrs. Clinton handled it, Mr. Herring wrote.

More than 100 other messages Mrs. Clinton handled contained information that was classified at the time she sent or received it, even though it had no markings at all, Mr. Comey said. Thousands of additional emails were deemed classified after the fact.

Congressional Republicans have questioned the FBI’s conclusions and demanded access to the investigative files, which the FBI turned over Tuesday. Among those were the notes of FBI agents’ interview with Mrs. Clinton, which came just before Mr. Comey recommended clearing her of charges.

Democrats said releasing the documents to Congress set a bad precedent and feared the information would be leaked to damage Mrs. Clinton in her presidential campaign.

They also said raising questions about the investigation is an attempt to keep the email issue alive.

“The FBI already determined unanimously that there is insufficient evidence of criminal wrongdoing,” Mr. Cummings said. “Republicans are now investigating the investigator in a desperate attempt to resuscitate this issue, keep it in the headlines and distract from Donald Trump’s sagging poll numbers.”

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