Mr Fernandes tweeted he had "strong information" the plane's "black box" flight recorders had been found.
There has been no confirmation, but it comes amid an intense operation to locate the plane's main fuselage - believed to contain missing bodies.
QZ8501 disappeared in bad weather on 28 December with 162 people on board.
It was flying from Surabaya in Indonesia to Singapore.
Priority
Mr Fernandes, AirAsia's chief executive, tweeted on Sunday that he had been "led to believe" that the recorders had been found.
"Still not confirmed. But strong info coming. But my man [sic] thoughts is fuselage," he said.
Earlier he tweeted: "Let's hope today is a major breakthrough day and we can find main fuselage."
An official said earlier that a large object resembling the plane's body was found in a sonar scan of the search area in the Java Sea.
But Supriyadi, operations coordinator for Indonesia's search and rescue agency, later told the BBC's Indonesian service that reports that the fuselage had been found had not been confirmed.
Officials hope the "black box" flight recorders will be near the object - close to where the tail was found.
Supriyadi has said that if the body of the plane is found, the first priority of search teams would be to remove the remains of victims.
"Secondly we will search for the black box."A team of divers had been sent to investigate, he added, but poor weather conditions have once again been hampering the search efforts.
Search teams have also been hearing pings, believed to be from the aircraft's black boxes, near where the tail of the Airbus A320-200 aircraft tail was retrieved from on Saturday.
Rescue workers have been pulling bodies and wreckage from the sea but progress has been slow. Forty-eight bodies have been retrieved so far.
The cause of the crash is unknown but the plane had encountered bad weather and asked for a flight path change before communication was lost.
The "black box" flight data recorders are usually housed inside the rear part of the plane.
They are designed to survive a crash and being submerged in water, and contain underwater locator beacons which emit the so-called "pings" for at least 30 days.
Finding them has been one of the top priorities for search teams as they provide crucial clues from the last moments of the flight before it came down.
