WASHINGTON—The Central Intelligence Agency has begun moving weapons to Jordan from a network of secret warehouses and plans to start arming small groups of vetted Syrian rebels within a month, expanding U.S. support of moderate forces battling President Bashar al-Assad, according to diplomats and U.S. officials briefed on the plans.

The shipments, related training and a parallel push to mobilize arms deliveries from European and Arab allies are being timed to allow a concerted push by the rebels starting by early August, the...