WASHINGTON—A Russian cybersecurity firm whose products current and former U.S. officials suspect Moscow has used as a tool for spying was flagged by U.S. military intelligence as a potential security threat as early as 2004, according to new information the Defense Department provided to Congress.

In 2013, the Defense Intelligence Agency, the U.S. military spy service, also issued a Pentagon-wide threat assessment about products made by the company, Kaspersky Lab, according to an email this week from the Pentagon to the House...