One of Hillary Clinton’s top deputies at the State Department tried to “pressure the FBI” to hush up some of former Secretary Hillary Clinton’s classified emails, two FBI employees told investigators in files released Monday.

Patrick Kennedy, the undersecretary for management, considered a “quid pro quo” where he would let more FBI agents into Iraq in exchange for the FBI erasing an earlier judgment that had deemed some of the Clinton email messages secret, according to investigative notes of interviews agents had with the two employees.

The FBI files also show Mrs. Clinton’s team of lawyers and advisers may have exposed her classified emails to the internet after she left office. A half-dozen laptops were used to go over Mrs. Clinton’s messages and decide which ones to belatedly comply with the law and turn back over to the government, and at least one of them was subsequently hooked up to the internet, lawyers admitted to the FBI.

The Obama administration was indignant at the suggestion of bribery concerning Mrs. Clinton’s messages, and a State Department spokesman said it was “insulting” to think they would be part of such deal-making.

But coupled with hacked emails showing what appears to be information-sharing between the State and Justice departments and officials on Mrs. Clinton’s presidential campaign, the new FBI reports fueled GOP fears of a cover-up to aide the Democratic presidential nominee.

“A senior State Department official’s attempt to pressure the FBI to hide the extent of this mishandling bears all the signs of a cover-up,” House Speaker Paul D. Ryan said in a statement after the FBI released its third batch of investigative files. “This is why our aggressive oversight work in the House is so important, and it will continue.”

GOP presidential nominee Donald Trump’s campaign said the new information was “undeniable proof” of collusion between Mrs. Clinton’s operation and the FBI and the Justice and State departments.

And the new revelations also provided a rallying point for a GOP that’s been fractured in recent weeks, with Mr. Trump and his critics within his party all agreeing that Mr. Kennedy should be excused from his job.

Mark Toner, a State Department spokesman, said that wasn’t going to happen, adding that the longtime diplomat still has the full backing of current Secretary of State John F. Kerry.

Mr. Kennedy has been a major figure in Mrs. Clinton’s political troubles over the last few years. He was also at the center of the 2012 Benghazi terrorist attack, though the State Department’s official report on that debacle blamed Mr. Kennedy’s underlings for the lax security around the diplomatic compound.

More recently, Mr. Kennedy has been implicated in the mishandling of Mrs. Clinton’s emails. He communicated with her via her secret account, but said in sworn testimony earlier this year that he didn’t recall paying attention to what address she was using, and didn’t worry at the time about whether she was complying with federal records laws and department policies.

Mr. Kennedy was the State Department employee who initially received the work application of Bryan Pagliano, the Clinton campaign staffer who went on to work for both the State Department and the Clinton family, including managing the secret email server.

“This guy has nine lives,” a former diplomat told Foreign Policy in a profile of Mr. Kennedy this summer.

In the latest documents, a person whose name was redacted but who appears to be an FBI employee said Mr. Kennedy was not satisfied with the FBI’s determination that some of Mrs. Clinton’s messages were classified. Mr. Kennedy pressured the law enforcement agency to change its ratings, and offered an exchange that the State Department “would reciprocate by allowing the FBI to place more agents in countries where they are presently forbidden,” the accuser told FBI agents.

Mr. Kennedy later presided over a meeting where he asked if the FBI could “see their way to marking the email unclassified.” Mr. Kennedy then went up the chain of command at the FBI, attempting to get the classified emails marked as unclassified, but was rebuffed at each step.

Another FBI employee said Mr. Kennedy contacted him last year, as the Clinton emails were being processed, and also sought to lower the classification of secret emails. That employee said Mr. Kennedy suggested marking the messages under a nonclassified exemption that would still allow the information to be kept private, but without the controversial stamp of secret material that has proved embarrassing to Mrs. Clinton.

“Kennedy … stated that the ‘B9’ classification would allow him to archive the document in the basement of [the State Department] never to be seen again,” the FBI said in its investigative notes.

Mr. Kennedy again offered an exchange, saying he would help the FBI get more agents into Iraq, the investigative file says. The FBI refused the offers and did not alter its classification, the file says.

The State Department said Monday that it did accept the FBI’s classification judgment, and also said “no increase in FBI Iraq slots resulted” from the quid pro quo offer.

“There’s no quid pro quo. Absolutely not,” Mr. Toner, the department spokesman, said. “There was a discussion over the classification of this material.”

The spokesman said it was the FBI employee, not the State Department, that raised the issue of additional agents in Iraq.

The White House also denied suggestions of any deal-making, with press secretary Josh Earnest waving those reports aside at his press briefing Monday.

When a reporter suggested that both the State Department and the FBI could be expected to deny the unflattering story, Mr. Earnest responded, “You’re suggesting that they would lie? Look, man, if that’s the place that we’re going to be in, then I think maybe you should go ask somebody else.”

He said the matter concluded without the FBI prosecuting anyone, and he blamed House Oversight Committee Chairman Jason Chaffetz, Utah Republican, for promoting a false story.

“I recognize that Congressman Chaffetz likes to try to make a big deal of these kinds of things,” Mr. Earnest said. “This is a member of Congress who is leading a Benghazi commission that Republicans in Congress acknowledged was geared solely to try to drive down Hillary Clinton’s poll numbers.”

He also criticized Mr. Chaffetz for previously handing out a business card with a gmail address on it, calling him “the worst kind of messenger” for discussing proper email habits.

• Dave Boyer contributed to this article.

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