The Airbus A320 went missing over the eastern Mediterranean, soon after entering Egyptian airspace.
The Egyptian military has denied a report from EgyptAir that a distress signal was sent by the plane.
There were 56 passengers – including three children – seven crew members and three security personnel on board Flight MS804, the airline said.
The airline said the passengers included 30 Egyptians, 15 French citizens, one Briton, two Iraqis, as well as people from Canada, Belgium, Kuwait, Saudi Arabia, Algeria, Sudan, Chad and Portugal.
Both the Greek and Egyptian armed forces are involved in the search for the plane. France says it is sending boats and planes to help in the operation.
Flight MS804 left Paris’ Charles de Gaulle airport at 23:09 local time on Wednesday (21:09 GMT) and was scheduled to arrive in the Egyptian capital soon after 03:15 local time on Thursday.
It was flying at 37,000ft (11,300m) over the eastern Mediterranean when contact was lost, at 02:45 Cairo time (00:45 GMT).
In what is thought to have been the last known contact with the plane, Greek air traffic controllers spoke to the pilot over the island of Kea and he “did not mention any problems”, Kostas Litzerakis of Greece’s civil aviation department told Reuters news agency.
Egypt’s state-run newspaper al-Ahram quoted an EgyptAir statement as saying the Egyptian army’s rescue and search had received a distress call from the plane at 04:26 local time – which would be around two hours after the flight disappeared.
But the Egypt’s military subsequently said that no such signal was received.
Flightradar24 listed details of the plane’s journey on Wednesday which showed it had flown from Asmara, in Eritrea, to Cairo, then on to Tunis, in Tunisia, before heading, via Cairo, to Paris.
Egyptian Prime Minister Sherif Ismail has arrived at the airport in Cairo, along with the families of those on board, state-run Nile News TV reports.
Aviation analyst Alex Macheras told the BBC that Airbus A320s were regularly used for short-haul budget flights and had “an amazing safety record”.
In March, an EgyptAir plane was hijacked and diverted to Cyprus. The attacker later surrendered and all hostages were released.
Last October, a Russian passenger plane flying from Sharm el-Sheikh crashed over the Sinai peninsula killing all 224 people on board. Officials in Moscow and Egypt later said the aircraft was brought down by an explosive device.
Islamic State militants said they had bombed the plane.