The e-mails appear to come from an fbi.gov address. They tell recipients that they have accessed illegal Web sites and that their Internet use has been monitored by the FBI's "Internet Fraud Complaint Center," the FBI said.
The messages then direct recipients to open an attachment and answer questions. The computer virus is in the attachment.
"Recipients of this or similar solicitations should know that the FBI does not engage in the practice of sending unsolicited e-mails to the public in this manner," the FBI said in a statement.
The bureau is investigating the phony e-mails.
The agency earlier this month shut down fbi.gov accounts, used to communicate with the public, because of a security breach. A spokeswoman said the two incidents appear to be unrelated.