U.S. airlines and their unions have joined forces to push the Federal Aviation Administration to let pilots do what was once unthinkable: sleep on the job.

Though the practice of nodding off midflight in the cockpit is now strictly forbidden by the FAA, U.S. airlines and pilot unions say there is reputable research supporting the notion that so-called controlled napping can enhance safety by making crews more alert during critical, often hectic descents and landings.

For...