A Washington, D.C. -based grand jury issued "a broad subpoena dated March 11" to at least one person involved in the development of the aircraft, the journal reported, citing people familiar with the matter.
Experts from the US National Transportation Safety Board and the plane's manufacturer Boeing are among those involved in the investigation.
Under global rules, a preliminary report on the crash must be released within 30 days.
But because of budget constraints, the FAA delegated aspects of the approval process to Boeing itself, according to sources.
"Boeing does not respond to or comment on questions concerning legal matters, whether internal, litigation, or governmental inquiries", Boeing spokesman Charles Bickers said in an email.
Both the flight data recorder and the cockpit voice recorder were sent to Paris for analysis by the French air accident investigation agency BEA.
Victims rise to 50 in Christchurch mosque attacks, 1 suspect left
Rashid, from Abbottabad, Pakistan, and a New Zealand resident for nine years, was in the mosque with his 21-year-old son. Huge piles of flowers were laid at sites near the mosques and crowds of people of all faiths gathered to pay respects.
Moges said that they will release more information within one month and a detailed report is to be expected.
Suspicions emerged that faulty sensors and software may have contributed to the two crashes in less than six months.
In both cases flight tracking data showed the aircraft's altitude had fluctuated sharply, as the planes seemed to experience erratic climbs and descents.
The pilot of the Indonesian flight is also reported to have attempted to return to the airport after reporting flight control problems, which is also similar to the Ethiopian flight.
Boeing said it supports the grounding of its planes as a precautionary step, while reiterating "full confidence" in their safety.
It prevents a jet from pointing upwards at too high an angle, where it could lose its lift.
Boeing is facing scrutiny over the Maneuvering Characteristics Augmentation System (MCAS), a software program that was created to help its new planes avoid stalling in mid-air.
"A Boeing spokesman said "(the) 737 MAX was certified in accordance with the identical FAA requirements and processes that have governed certification of all previous new airplanes and derivatives", according to the Reuters news agency. He called that amount of training unacceptable.