Boeing had said it would submit a fix to the FAA last week, and it had gathered hundreds of industry representatives at its Seattle area facilities last Wednesday to demonstrate software changes.
But the FAA said Monday that the company concluded "additional work" is needed.
"The FAA expects to receive Boeing's final package of its software enhancements over the next few weeks for FAA approval," the agency said in a statement. "Boeing is in the process of continuous review of the 737 MAX Flight Control System to ensure that Boeing has identified and adequately addressed all relevant issues."
Boeing acknowledged the new timeline, saying that the software change would be "completed in the coming weeks." It did not say why the timeline had changed.
"The safety is our first priority, and we will take a thorough and methodical approach to the development and testing of the update to ensure we take the time to get it right," the company said.
American Airlines said Monday afternoon that it was aware of MAX "may be delayed" in returning to service. It has been forced to cancel dozens of flights daily while the planes are waiting for a fix.
Capt. Dennis Tajer of the Allied Pilots Association, which represents the American Airlines pilot, said the group was encouraged by Boeing's recent "sense of energy" to produce a fix but also realized that the timeline was ambitious.
"We were disregarding that because having the pilot fully engaged, as they are, with all the information and knowing what the training will be all, put that out as more of a distant timeline for getting the aircraft flying," Tajer said CNN
The FAA followed the other countries' aviation authorities in grounding the MAX on March 13 following the second crash in less than five months. The agency said that it had reviewed satellite data and physical evidence that showed similarities between the Lion Air flight that crashed in October and the flight of Ethiopian Airlines that crashed days earlier than March.
A preliminary report on the Ethiopian accident is expected in the coming days.
Software designers, the Boeing engineers and the pilot pilot had worked out a set of changes to a stabilization system known as MCAS, which has been under scrutiny by investigators while joining the Lion Air crash, according to a company official. The fix brings in multiple layers of protection and is accompanied by a training regimen.
Tajer, the Allied Pilots Association, said the software changes Boeing demonstrated last week "directly addressed the concerns that we voiced to Boeing" following the October crash.
The FAA statement says the software changes would require a "rigorous safety review."
"The FAA will not approve the software for installation until the FAA is satisfied with the submission," the agency said.
CNN's David Shortell contributed to this report.