"He did clarify with the passengers that he was wishing the mother on board a happy birthday."
The spokesman added that the air traffic controller had been working out of the Terminal Radar Approach in Westbury, Long Island, when he contacted the pilot and requested that he send a birthday message to his mother, who was on the plane.
In a statement, the Federal Aviation Administration said: "Pilots and controllers will sometimes engage in brief greetings.
"If such conversations go beyond this limit, controllers are counselled to refrain from such unnecessary talk."
The FAA has launched a review into the air traffic controller's request. The pilot, who reported the incident to the authorities at Southwest, is not under investigation.