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The FAA said it "is reviewing this information to determine what action is appropriate."
A person briefed on the matter said Boeing failed to turn over the documents to the FAA for four months and that the Justice Department is also in possession of the messages.
Boeing said on Friday Muilenburg had called FAA Administrator Steve Dickson to respond to the concerns raised in his letter and assured him that the company "is taking every step possible to safely return the Max to service."
It said it had produced the document containing a former Boeing employee's statements to the appropriate investigating authority earlier this year, and brought it to the attention of the Department of Transportation on Thursday.
Boeing has been cooperating with the House of Representatives Transportation & Infrastructure Committee's probe into the 737 Max and will continue to do so in investigations by US authorities, the company said.
Boeing turned the documents over to the US Department of Justice and the FBI in February, one source said.
Federal prosecutors aided by the FBI, the Department of Transportation's inspector general and several blue-ribbon panels are investigating the 737 Max's certification. And the US Senate Commerce Committee confirmed it will question Muilenburg at an Oct. 29 hearing, one day before a House of Representatives panel is scheduled to question him.
The 737 Max has been grounded worldwide since March, forcing more than 100 daily flight cancellations at large US operators of the aircraft like Southwest Airlines Co and American Airlines, and eroding their profits.
Boeing has said the grounding has already cost it at least $8 billion.
'I basically lied'
Forkner responded soon after: "Granted I suck at flying, but even this was egregious." At one point Forkner said "there are still some real fundamental issues" in the simulator.
"If you read the whole chat, it is obvious that there was no 'lie,'" Forkner's lawyer David Gerger said by email on Friday.
"The simulator was not reading right and had to be fixed to fly like the real plane. Mark's career - at Air Force, at FAA, and at Boeing - was about safety. And based on everything he knew, he absolutely thought this plane was safe."
Forkner is no longer employed by Boeing. The Seattle Times reported in September that he repeatedly invoked his Fifth Amendment right to not turn over documents subpoenaed by the Justice Department.
Boeing has been revising the 737 Max software as part of its efforts to win fresh approval for the jet to fly again.
Southwest, the world's largest operators of the 737 Max, said in a statement it had been unaware of the messages but continued to work with the FAA and Boeing to safely return the aircraft to service.
The Southwest pilots union, which has filed a lawsuit against Boeing for lost wages during the grounding, said in a statement that the document was "more evidence that Boeing misled pilots, government regulators and other aviation experts about the safety of the 737 Max."
"As pilots, we have to be able to trust Boeing to truthfully disclose the information we need to safely operate our aircraft. In the case of the 737 Max, that absolutely did not happen," Southwest Airlines Pilots Association President Jon Weaks said.
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