The department is now reviewing some 55,000 pages of emails. A first batch of 3,000 pages was made public on June 30.

In the course of the email review, State Department officials determined that some information in the messages should be retroactively classified. In the 3,000 pages that were released, for example, portions of two dozen emails were redacted because they were upgraded to “classified status.” But none of those were marked as classified at the time Mrs. Clinton handled them.

In a second memo to Mr. Kennedy, sent on July 17, the inspectors general said that at least one email made public by the State Department contained classified information. The inspectors general did not identify the email or reveal its substance.

The memos were provided to The New York Times by a senior government official.

The inspectors general also criticized the State Department for its handling of sensitive information, particularly its reliance on retired senior Foreign Service officers to decide if information should be classified, and for not consulting with the intelligence agencies about its determinations.

In March, Mrs. Clinton insisted that she was careful in her handling of information on her private account. “I did not email any classified material to anyone on my email,” she said. “There is no classified material. So I’m certainly well aware of the classification requirements and did not send classified material.”

In May, the F.B.I. asked the State Department to classify a section of Mrs. Clinton’s emails that related to suspects who may have been arrested in connection with the 2012 attacks in Benghazi, Libya. The information was not classified at the time Mrs. Clinton received it.