Neither Southwest nor its pilots' union responded immediately to requests for comment.
The FAA investigation was disclosed Wednesday by the US Office of Special Counsel, which represents federal whistleblowers. It reported another accident — a plane that went off the runway in Burbank, California — in which an FAA review was "fast-tracked" under pressure from Southwest.
In a separate review of that incident, the National Transportation Review Board found that the co-pilot's conduct was "highly unprofessional."
The FAA said it agreed with some of the allegations raised by three whistleblowers and took corrective steps including more training and development of audits to ensure compliance with aviation-safety guidelines.
The special counsel's office investigated eight allegations raised by four whistleblowers. The office forwarded its findings to the White House and Congress.
In a response to the special counsel's office, the FAA said it found "mismanagement and lack of oversight" by the office monitoring Southwest that has persisted despite management and staff changes over the years. The FAA said new executives will provide "a fresh opportunity to evaluate" the oversight of Southwest.
Some of the whistleblower accusations have been leveled before, including Southwest's use of planes bought overseas without verifying their maintenance and inspection records.
In more than half of those cases, the airline found that the planes had undergone repair work that wasn't documented or couldn't be verified.
The FAA said some of the whistleblower allegations couldn't be proven, including a claim that Southwest routinely assigns too much work for mechanics to handle.
A separate allegation that the FAA improperly certified Southwest for long overwater flights — approval the airline needed to sell flights to Hawaii — was examined by the Transportation Department's inspector general, who could not verify the claim.
An FAA spokeswoman said the agency takes the special counsel's concerns seriously and has adopted its recommendations.
Dallas-based Southwest has faced questions about safety over the years and has paid millions to settle safety violations, however, it has a good record over its five-decade history.
No passenger had died in an accident involving a Southwest plane until 2018 when a woman was killed after an engine broke apart over Pennsylvania and debris shattered the window next to her seat.
In 2005, a 6-year-old boy riding in a car was killed when a Southwest jet landing in Chicago skidded off the runway, crashed through a fence and hit the car.