In a tense, private meeting last week in Washington, the head of the Federal Aviation Administration reprimanded Boeing’s chief executive for putting pressure on the agency to move faster in approving the return of the company’s 737 Max jet.

This was the first face-to-face encounter between the F.A.A. chief, Stephen Dickson, and the executive, Dennis A. Muilenburg, and Mr. Dickson told him not to ask for any favors during the discussion. He said Boeing should focus on providing all the documents needed to fully describe the plane’s software changes, according to two people briefed on the meeting.

It was a rare dressing-down for the leader of one of the world’s biggest companies, and a sign of the deteriorating relationship between Mr. Muilenburg and the regulator that will determine when Boeing’s most important plane will fly again.

The global grounding of the 737 Max has entered its 10th month, after two crashes that killed 346 people, and the most significant crisis in Boeing’s history has no end in sight. Mr. Muilenburg is under immense pressure to achieve two distinct goals. He wants to return the Max to service as soon as possible, relieving the pressure on Boeing, airlines and suppliers. Yet the company and regulators must fix an automated system known as MCAS found to have played a role in both crashes, ensuring the Max is certified safely and transparently. Caught in the middle, Mr. Muilenburg has found himself promising more than he can deliver.