WASHINGTON—Syria has begun moving parts of its vast arsenal of chemical weapons out of storage facilities, U.S. officials said, in a development that has alarmed many in Washington.

The country's undeclared stockpiles of sarin nerve agent, mustard gas and cyanide have long worried U.S. officials and their allies in the region, who have watched anxiously amid the conflict in Syria for any change in the status or location of the weapons.

American...