Ihab Raslan, a spokesman for the Egyptian civil aviation authority, said the plane was about to enter Egyptian airspace when it disappeared from radar.
The airline, however, said the Airbus A320 had vanished 10 miles after it entered Egyptian airspace.
Around 15 family members of passengers on board the missing flight arrived at Cairo airport and authorities brought doctors to the scene after several collapsed.
EgyptAir Flight 804 was lost from radar at 2.45am local time when it was flying at 37,000 feet, the airline said.
Egyptian armed forces were searching for the plane, which was carrying 56 passengers, including one child and two babies, and 10 crew. The pilot had 6,000 flight hours’ experience.
Airbus said it was aware of the disappearance, but “we have no official information at this stage of the certitude of an accident”, company spokesman Jacques Rocca said.
Greece joined the search and rescue operation for the EgyptAir flight with two aircraft, one C-130 and one early warning aircraft, the Hellenic National Defence General Staff said.
They said one frigate was also heading to the area and helicopters were on standby on the southern island of Karpathos for potential rescue or recovery operations.
Ahram, Egypt’s state-run newspaper, quoted an airport official as saying that the pilot had not sent a distress signal before it disappeared.
The last contact with the plane was 10 minutes before it vanished, he was quoted as saying.
EgyptAir said the Airbus A320 was carrying 66 people on a flight from Paris to Cairo when disappeared from the radar at 2:45 a.m. local time.
An EgyptAir plane was hijacked and diverted to Cyprus in March. A man who admitted to the hijacking, described by Cypriot authorities as “psychologically unstable”, is in custody in Cyprus.
The incident renewed security concerns months after a Russian passenger plane was blown out of the sky over the Sinai Peninsula. The Russian plane crashed in Sinai on October 31, killing all 224 people on board.
Moscow said it was brought down by an explosive device and a local branch of the extremist Islamic State group said it planted it.
In 1999, EgyptAir Flight 1990 crashed into the Atlantic Ocean near the Massachusetts island of Nantucket, killing all 217 people aboard.
US investigators filed a final report that concluded its co-pilot switched off the autopilot and pointed the Boeing 767 downward.
But Egyptian officials rejected the notion of suicide altogether, insisting some mechanical reason caused the crash.
Reporters gathered in front of the small, empty EgyptAir counter at Terminal 1 of Charles de Gaulle Airport. Staff said EgyptAir employees were on their way.