Boeing CEO Dennis Muilenburg reportedly will decline his management bonus for 2019 as the company continues to deal with the fallout from two deadly crashes of the company's 737 Max aircraft.

Boeing Chairman David Calhoun told CNBC that Muilenburg made the suggestion to him on Saturday morning. The call was made "with the purpose of suggesting that he not take any compensation for 2019 in the form of bonuses, which of course is most of your compensation [as a chief executive]," Calhoun told the network.

However, Boeing had recently told employees in an internal message that the company would not pay any bonuses for 2019.

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Muilenburg made $23 million in 2018, the year of the first Boeing 737 Max crash, minutes after takeoff into the sea off Indonesia. Only $1.7 million of his total compensation for that year was paid through his salary, according to Boeing's proxy statement. He earned about $13.1 million from a bonus tied to individual and company performance, and another $7.3 million in stock awards.

A second 737 Max crash in March 2019, in Ethiopia, also minutes after takeoff, changed everything for Boeing, its airline customers and the flying public. The accident led to a worldwide grounding of the jets that's expected to continue into 2020, and perhaps beyond.

Calhoun told CNBC that his Saturday compensation discussion with Muilenburg "came in two fronts: one, no short-, no long-term bonus, and three, no consideration for [stock] equity grants until the Max in its entirety is back in the air and flying safely."

He said the 737 Max might not return to service "in its entirety" until 2021.

Safety concerns around the 737 Max have battered the company's financial results, with the aircraft maker in July announcing its worst quarter in the company's history. Despite that turbulence, Boeing shares have risen nearly 9% this year.