Dallas police is currently leading the investigation into the Friday night incident.
Its officers responded to reports of gunfire at 21:48 local time (03:48 GMT on Saturday) and arrived to find that the plane had been struck.
The Federal Aviation Administration said in a statement that the bullet hit near the cockpit of the Boeing 737-800 Max aircraft.
No details have formally been released on the circumstances surrounding the incident, but local TV station WFAA reported that police believed the shooting was a result of random gunfire.
The outlet confirms 99 people were aboard the flight.
Southwest has said it will accommodate all passengers on other flights.
Earlier this week, three planes that were either landing or taking off from the main airport in Port-au-Prince, the capital of Haiti, were struck by gunfire as gang violence ravages the Caribbean nation.
Those incidents prompted the Federal Aviation Administration to issue a 30-day ban on US airlines flying to Haiti.