American officials will not discuss what classified information the investigators found in Ms. Raphel’s home. The current and former American officials who discussed the case did so on the condition of anonymity because they were not authorized to talk about it publicly.

Over the years, the stories of American officials mishandling classified information have at times seemed as peculiar as they were serious. John P. O’Neill, a counterterrorism specialist for the F.B.I., once lost a briefcase full of government secrets in a Florida hotel. Samuel R. Berger, the national security adviser to President Bill Clinton, stole classified documents from the National Archives and hid them under a construction trailer. As attorney general, Alberto R. Gonzales took material about the nation’s warrantless wiretapping program home with him.

One C.I.A. director, John M. Deutch, stored classified information on his home computer. Another C.I.A. director, David H. Petraeus, shared his highly classified journals with his mistress, then lied to the F.B.I. about it. Hillary Rodham Clinton used a private email system when she was secretary of state that investigators say contained classified information, although Mrs. Clinton and the State Department say the information was not marked as classified.

The punishment for mishandling classified information has varied wildly. Mrs. Clinton has not been charged with wrongdoing. Mr. Berger pleaded guilty to a misdemeanor. Mr. Deutch received a pardon from Mr. Clinton and was never charged. Mr. Gonzales and Mr. O’Neill were not charged. In the most recent case, the Justice Department allowed Mr. Petraeus to plead guilty to a misdemeanor, despite strong objections from investigators. That deal was so contentious that the F.B.I. director, James B. Comey, personally appealed to the attorney general, Eric H. Holder Jr., and said that Mr. Petraeus’s crimes warranted felony charges, according to two government officials involved in the case. F.B.I. agents are still angry about that decision and say it set a standard that will make it harder to bring cases in the future.

In discussions with prosecutors, according to several government officials, Ms. Raphel and her lawyer have cited the Petraeus case as the vital precedent. If passing secrets — including notes on war strategy and the names of covert officers, which Mr. Petraeus shared — and lying about it amount to a misdemeanor, then, Ms. Raphel says, she should not face any charges.

Some American investigators remain suspicious of Ms. Raphel and are loath to abandon the case entirely. Even if the government cannot mount a case for outright spying, they are pushing for a felony charge related to the classified information in her home. Several officials acknowledged, however, that the case would be difficult to prosecute because it would require intelligence agencies to declassify information and would probably reveal secrets about American surveillance of foreign officials.