"Although there was never a quid pro quo, these allegations were nonetheless referred to the appropriate officials for review," the FBI said Monday in a statement.
The State Department said Kennedy had been trying to understand the FBI's classification decisions.
"This allegation is inaccurate and does not align with the facts," department spokesman Mark Toner said in a statement. He also said there was never an increase in the number of FBI agents assigned to Iraq as a result of the conversations.
The disclosure was included in 100 pages the FBI released from its now-closed investigation into whether the former secretary of state and her aides mishandled sensitive government information that flowed through the private mail server located in her New York home.
Kennedy proposed using an obscure provision under the U.S. Freedom of Information Act to keep the existence of the secret document from becoming public, the FBI files said.
The provision, known as "B9," is intended to protect geological and geophysical information and data, including maps, concerning wells and is the most rarely used FOIA exemption. In fiscal 2015, the Obama administration cited it only 46 times out of 769,903 information requests. Most of those cases involved the Environmental Protection Agency and the Defense Department.
"Kennedy told (redacted) that the FBI's classification of the email in question caused problems for Kennedy and Kennedy wanted to classify the document as 'B9,'" The FBI report says. "Kennedy further stated that the 'B9' classification would allow him to archive the document in the basement of DoS (Department of State) never to be seen again."
In the end, the FBI did not change the classification level.
The FBI official said that after learning later that the information in question concerned the Benghazi attacks, he contacted Kennedy and told him there was "no way he could assist" with declassifying the material found in the email.
The Associated Press reported the existence of the secret Benghazi-related email in May 2015, though the classified content of the document has never been made public.
At the time, administration officials acknowledged interagency disagreements about whether certain information in the emails was classified, disagreements that the State Department said were contributing to the slow processing of their public release under the Freedom of Information Act.
Jason Miller, a spokesman for Republican nominee Donald Trump's campaign, said the FBI's summary of the conversation "should be distressing for anyone who cares about keeping classified information safe."
"The Clinton campaign must immediately disavow Kennedy and give the American people a full accounting of any communication relating to the request to change the classification levels on certain emails," Miller said.
The Clinton campaign did not immediately respond to a request for comment from AP.