An Egyptian aircraft spotted two orange objects near the Mediterranean island of Karpathos, which are believed to have come from EgyptAir flight MS804, Greek military officials said.
The flight from Paris to Cairo on Thursday morning disappeared from radar 10 miles inside Egyptian air space at 2.30am Cairo time (1.30am BST) after taking off just under three-and-a-half hours earlier from Charles de Gaulle Airport.
The news of debris came as Egyptian civil aviation minister Sherif Fathy told a Cairo press conference: "If you analyse the situation properly the possibility of ... having a terror attack is higher than the possibility of having a technical (failure)."
A Briton was among the passengers aboard the flight, which made a series of sharp turns and plummeted thousands of feet through the air before disappearing.
A major search and rescue operation was launched after the aircraft came down, reportedly near Karpathos, which is around 50 miles east of Crete in the eastern Mediterranean.
Its loss was "most likely a terrorist attack", Alexander Bortnikov, director of the Russian Federal Security Service (FSB), told news agencies.
Panos Kammenos, the Greek defence minister, told reporters in Athens: "It turned 90 degrees left and then a 360-degree turn toward the right, dropping from 38,000 to 15,000 feet and then it was lost at about 10,000 feet."
Among those on board were a child and two babies, EgyptAir said.
The airline said the 56 passengers included 30 Egyptians, 15 French, two Iraqis and one each from Britain, Sudan, Chad, Portugal, Algeria, Canada, Belgium, Kuwait and Saudi Arabia.
French president Francois Hollande earlier confirmed the aircraft had come down, saying: "I have been informed that the aircraft that left Paris to go to Cairo has been lost. It crashed."
He added that "no hypothesis is excluded" in relation to the cause.
Foreign Secretary Philip Hammond said: "We know that one British passport holder boarded the flight in Paris and our staff are providing support and assistance to the family at this difficult time. We will continue to help in any way we can."
The Airbus A320 was built in 2003 and was flying at 37,000ft, the airline said on Twitter.
It tweeted that the pilot had logged 6,275 flying hours, including 2,101 hours on the A320, and the co-pilot had logged 2,766 hours.
Flight MS804 left Charles de Gaulle Airport at 11.09pm local time (10.09pm BST).
French foreign minister Jean-Marc Ayrault said France had offered military assistance to Egypt to help with the search.
There was confusion over whether a distress signal had been sent by the Airbus A320.
Egypt's civil aviation authority said one was received at 4.26am local time, believed to be an automated message rather than one sent by the pilot.
However the Egyptian military later said it had received no distress message from the aircraft, in a statement on its website.
In March, a domestic EgyptAir flight with 72 passengers on board had to make an emergency diversion to Larnaca, Cyprus, after an alleged hijacking.
In October last year, 224 people were killed when a Russian aircraft crashed over Egypt's Sinai Peninsula minutes after it took off from the Red Sea resort of Sharm el-Sheikh.
EgyptAir has provided free contact numbers for families concerned for relatives. From outside Egypt, anyone concerned should call + 202 2598 9320.
A British ship and plane have been deployed to help in the search for traces of the missing flight.
The Royal Navy vessel RFA Lyme Bay, which was operating south-east of Crete, was due to arrive in the vicinity of the plane's last known location late on Thursday, Downing Street said.
It is planned that the ship, with 60 personnel on board, will search along the track of the airliner's route.
Meanwhile, an RAF C130 aircraft was being deployed from the UK air base at Akrotiri, Cyprus. Equipped with surveillance kit, radar, high-powered binoculars and night vision goggles, it has the capability for day and night searching in all weathers.
The UK Government is treating the incident as an "important priority", with officials from a number of Whitehall departments working on it throughout the day and ministers briefed when necessary, said Number 10.
Defence Secretary Michael Fallon said: "Our thoughts remain with the families of those on board EgyptAir flight MS804 as they await further information.
"To support the ongoing search efforts, I have today directed RFA Lyme Bay to the area, and I have also offered a C-130 Hercules aircraft from RAF Akrotiri to support the Egyptian-led effort.
"We stand ready to offer further assistance should it be required."