WASHINGTON—Nearly a dozen American troops were injured in the Iranian missile attack on two bases in Iraq last week, Defense Department officials said, after initially stating that there were no casualties in the strikes.

Eleven individuals are being screened for traumatic brain injuries following the attacks on two bases in Iraq that house American troops. Iran fired a dozen rockets total at Erbil in northern Iraq and the sprawling Al Asad air base in the west in retaliation for the American killing of Maj. Gen. Qassem Soleimani...