WASHINGTON—A highly secure computer system where aides to President Trump reportedly stashed the details of his call with Ukraine’s leader is so secretive that even top White House national-security aides don’t have regular access, according to former officials familiar with its operation.

It is the most tightly controlled of at least four different computer systems used by the National Security Council staff, they said, and contains the most precious of American secrets: U.S. covert actions in other countries and counterintelligence...