The state-run Xinhua news agency had reported the compensation claim earlier on Wednesday, but did not provide any financial or other details.
A China Eastern spokesman confirmed the report, but declined further comment.
On March 11, China became the first country to order its domestic air fleet to ground the plane as a result of the two tragedies.
The day before, an Ethiopian Airlines 737 MAX crashed minutes after takeoff from Addis Ababa, killing all 157 people on board, including eight from China.
That followed the October crash of a Lion Air 737 MAX, which went down shortly after takeoff from Jakarta, killing all 189 people on board.
China Eastern "said that grounding Boeing 737 MAX 8 planes had resulted in great losses for the company, and the losses are still expanding", Xinhua reported.
It said the airline declined to disclose details of a letter of claim that was sent to Boeing.
The Shanghai-based carrier has grounded 14 MAX planes while the US aviation giant addresses the safety risks.
Boeing acknowledged Saturday it had to correct flaws in its 737 MAX flight simulator software used to train pilots.
The firm's statement about the simulator marked a first acknowledgement of a shortcoming since the two accidents led to the worldwide grounding of the top-selling airliner.
The anti-stall system on the MAX models has been blamed in large part for the Ethiopian Airlines tragedy.