Some of the nation’s intelligence agencies raised alarms last spring as the State Department began releasing emails from Hillary Clinton’s private server, saying that a number of the messages contained information that should be classified “top secret.”

The diplomats saw things differently and pushed back at the spies.

In the months since, a battle has played out between the State Department and the intelligence agencies — as well as Congress — over what information on Mrs. Clinton’s private server was classified and what was the routine business of American diplomacy, according to government officials and letters obtained by The New York Times.