Election machines used in more than half of U.S. states carry a flaw disclosed more than a decade ago that makes them vulnerable to a cyberattack, according to a report to be delivered Thursday on Capitol Hill.

The issue was found in the widely used Model 650 high-speed ballot-counting machine made by Election Systems & Software LLC, the nation’s leading manufacturer of election equipment. It is one of about seven security problems in several models of voting equipment described in the report, which is based on research conducted...