The carrier told its annual shareholders meeting that it lost about $128 million in sales after Boeing's next-generation jet was grounded in January over a series of battery glitches.
"We have started negotiations with Boeing about an appropriate amount of compensation," ANA's Executive Vice President Kiyoshi Tonomoto told investors, according to the Nikkei business daily.
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JAL said it lost about 3.9 billion yen because of the problems.
The two airlines, which are the world's two biggest operators of the planes, have now put their Dreamliner fleets back into service following a four-month suspension as regulators carried out a investigation.
A global grounding order was issued in January after lithium-ion batteries overheated on two different jets, with one of them catching fire while the aircraft was parked.
Boeing has not been able to identify the root cause of the problems, but rolled out modifications it said would ensure they were safe.
Meanwhile, ANA's chief executive Shinichiro Ito said JAL continues to receive government support after a high-profile bankruptcy restructuring that saw the carrier re-listed in Tokyo last year.
"I have no choice but to say that the playing field is not level," the Nikkei quoted Ito as saying.
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