Should the F.B.I. find emails that it had never seen before containing potentially classified material, copies would have to be sent to other government agencies to determine their classification — an elaborate and lengthy process.

The emails belong to Huma Abedin, the aide to Mrs. Clinton. Agents discovered them on a laptop seized by the F.B.I. that belongs to Ms. Abedin’s estranged husband, Anthony D. Weiner, who is under investigation on allegations that he exchanged illicit text messages with a 15-year-old girl.

While the hunt for classified information is the bureau’s first priority, it is not the most significant issue for either Ms. Abedin or Mrs. Clinton. Investigators have already determined that Mrs. Clinton and her aides improperly sent classified information on her private email server. The Justice Department concluded, though, that it could not prove they did it intentionally, which would be a crime. Finding more classified information among Ms. Abedin’s emails would not immediately change that conclusion.

The agents will also focus on whether this new material contains any evidence that anyone may have tried to conceal these newly discovered emails or others from investigators, which could amount to a crime. Ms. Abedin has said she turned over all the work-related emails she knew about to the F.B.I. months ago and does not know how her emails ended up on Mr. Weiner’s laptop. Officials have said in the past that there is no indication that Ms. Abedin or Mrs. Clinton tried to conceal information from the authorities.