The devices record information about the speed, altitude and direction of the plane as well as flight crew conversations and could hold vital clues to the cause of the deadly accident.
The Detik News website reported that the black box had been found near the aircraft fragments at the depth of 30 meters (98.4 miles) below the surface of the water off the Indonesian north coast, where the plane crashed on Monday.
"That happened several times during the flight", he said.
Indonesia's biggest low-priced carrier announced this year it was buying 50 Boeing-737 MAX 10 jets for $6.24 billion.
The Lion Air flight JT610, which was carrying 189 people, plunged into the Java Sea on Monday.
Edward Sirait, the president of Lion Air, has said the company was aware of reports of problems on the flight on Sunday and that a technical issue was "resolved in accordance with the procedures released by the plane manufacturer".
In response to Monday's crash, the Australia's government has banned officials and contractors from flying on Lion Air pending results of the investigation.
Boeing Co. experts are expected to arrive Indonesia on Wednesday and Lion Air has said an "intense" internal investigation is underway in addition to the probe by safety regulators.
Lion's co-founder Rusdi Kirana, now Indonesia's ambassador to Malaysia, said he had met with passengers' families and called the crisis "exhausting and saddening for everybody". After narrowing the possible location, "I started digging and cleaning the debris until I finally found an orange object", he said, standing on the deck of a ship next to his diving mate.
The accident has also resurrected concerns about Indonesia's poor air safety record which until recently saw the country's carriers facing years-long bans from entering European Union and USA airspace.
Inspections have already been carried out on Lion Air's other Boeing 737 MAX 8 planes, preventing them from flying.
Shoes of passengers of Flight 610 were laid out at the Tanjung Priok port in Jakarta.
National Transportation Safety Committee investigator Ony Soeryo Wibowo addressed media in Jakarta following the discovery, explaining that the particular piece of equipment was not recovered in its entirety.
His wife Ningsy Ayorbaba, interviewed at a Jakarta police hospital where she had taken her three children for DNA tests to help with victim identification, said: "My husband sent that video".
The cause of the crash has yet to be established, but questions have been raised after Lion Air admitted the plane had an unspecified technical issue on a previous flight. Its aircraft have been involved in at least 15 safety incidents and it has been placed under tougher worldwide safety restrictions than other Indonesian airlines.