Although the FBI decided against declassifying the email's contents, the claim of interference added fuel to Republicans' belief.
This is according to records of interviews with FBI officials released on Monday.
The accusation against Patrick Kennedy, the State Department's most senior manager, appears in the latest release of interview summaries from the Federal Bureau of Investigation's year-long investigation into Clinton's sending and receiving classified government secrets via her unauthorized server.
Although the FBI decided against declassifying the email's contents, the claim of interference added fuel to Republicans' belief that officials in President Barack Obama's administration have sought to protect Clinton, a Democrat, from criminal liability as she seeks to succeed Obama in the Nov. 8 election.
The FBI recommended against bringing any charges in July and has defended the integrity of its investigation.
Clinton has said her decision to use a private server in her home for her work as the U.S. secretary of state from 2009 to 2013 was a mistake and has apologized.
One FBI official, whose name is redacted, told investigators that Kennedy repeatedly "pressured" the various officials at the FBI to declassify information in one of Clinton's emails. The email was about the deadly 2012 attack on a U.S. compound in Benghazi, Libya, and included information that originated from the FBI, which meant that the FBI had final say on whether it would remain classified.
A State Department spokesman, Mark Toner, said Kennedy was not pressuring the FBI but was just trying to understand better how the FBI's classification process worked.
The dispute began in the summer of 2015 as officials were busy reviewing the roughly 30,000 emails Clinton returned to the State Department ahead of their court-ordered public release in batches in 2015 and 2016.
The official said the State Department's office of legal counsel called him to question the FBI's ruling that the information was classified, but the FBI stood by its decision.
Soon after that call, one of the official's FBI colleagues received a call from Kennedy in which Kennedy "asked his assistance in altering the email's classification in exchange for a 'quid pro quo.'"
The FBI official said he also joined at least two discussions in which Kennedy "continued to pressure" the FBI about the email. The official said Kennedy appeared to be trying to protect Clinton by minimizing the appearance of classified information in emails from the server that Clinton used while she was the country's most senior diplomat.