ISLAMABAD—An American construction worker who was arrested with a 40-inch sword, a pistol and night-vision goggles in northwestern Pakistan told investigators Tuesday that he wanted to kill Osama bin Laden to avenge the 2001 terrorist attacks on the U.S.

Gary Brooks Faulkner, 50 years old, of Greeley, Colo., was caught by Pakistani police Monday in the remote Bumburat Valley near the border of Afghanistan's Nuristan province, where he apparently hoped to find Mr. bin Laden.

Police...