EgyptAir Flight 804 sent automated warning messages minutes before it went down indicating smoke was detected in the aircraft’s nose, investigators said, while they stepped up efforts to garner more clues to what caused the plane to plunge into the Mediterranean.

The messages were sent automatically from the plane by onboard systems immediately before all contact was lost, the French air accident office said Saturday, confirming a report Friday in The Wall Street Journal. The messages indicate the flight-control system had...