The No. 2 official at the Justice Department delivered a blunt message last month to Apple Inc. executives: New encryption technology that renders locked iPhones impervious to law enforcement would lead to tragedy. A child would die, he said, because police wouldn’t be able to scour a suspect’s phone, according to people who attended the meeting.

At issue is new technology that Apple, Google Inc. and others have put in place recently to make their devices more secure. The companies say their aim is to satisfy consumer demands...