BEIRUT—Syrian opposition figures said the U.S. military informed them over the weekend that American airstrikes against Islamic State would begin this week, advising the Free Syrian Army to prepare its forces while a covert arming program run by the U.S. Central Intelligence Agency provided fresh weapons.

The U.S. bombarded Islamic State-held towns and cities overnight Monday to provide support to the FSA and pave the way for the allied opposition to attempt to clear and hold territory held by the Islamist militants. So far,...