Security researchers say a little-known Israeli startup exploited previously unknown bugs in Apple Inc.’s smartphone software to help foreign governments spy on their citizens.

The researchers say the surveillance software was the work of NSO Group Technologies Ltd., which sells primarily to government agencies. The researchers, at Citizen Lab, a group that investigates surveillance technology, and at mobile-security firm Lookout Inc., say they discovered the software in a link sent earlier this month to the phone of Ahmed...