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.
"This is a trick trying to get you to open a file so that your computer can be infected with a virus," said CBSNews.com Tech Columnist Larry Magid, noting this familiar trick is called "social engineering." "The FBI never sends out unsolicited email to the public and no reputable organization would ever send out attached files to the public unless there was a pre-existing relationship and a reason to send attached files.
"It's becoming increasingly common for virus writers and scam artists to use bogus security alerts to trick people into opening an attachment or providing information," said Magid.
"It's never a good idea to open an attached file in email unless you're expecting it. That goes even if it appears to come from a reputable organization or even a friend," Magid noted. "It's very easy to fake an email return address so a message that appears to come from the FBI or anyone else could easily be from an imposter."