Police said they were treating it as a criminal incident and not a potential terrorist attack.
"There was a firearms incident. A white male was wounded then died at 5.34pm," a police spokeswoman told the Reuters news agency.
Airport sources said the incident took place close to the head office of British Airways. Police said the shoot-out took place in Saxon Way off the A4.
Security at the airport was stepped up in February amid fears it might become a target for terrorist attacks.
