In an announcement Tuesday, the FBI advised computer users that the agency never sends unsolicited e-mails and that they should not open the attachments in the fraudulent messages.

The scam e-mail tells recipients that their Internet use has been monitored by the FBI and that they have accessed illegal Web sites, according to the FBI. The e-mails, which are sent from e-mail addresses including mail@fbi.gov, post@fbi.gov and admin@fbi.gov, then direct recipients to open an attachment and answer questions.

The text of the fake e-mails says: 'We have logged your IP-address on more than 30 illegal Websites. Important: Please answer our questions! The list of questions are attached.' The messages then include a fake name of an FBI official and the real address and phone number of the agency.

Catherine Milhoan, an FBI spokeswoman, said that recipients of the hoax e-mails began calling the FBI's Internet Crime Complaint Center yesterday to ask if they were legitimate. About 4,000 calls were received by last night, with some callers saying they had already clicked on the attachment. Others were cautious and wanted to check with the FBI before opening the attachment.

An investigation into the incident is continuing.