The employee is accused of passing details of eight high-profile customers including actress Sienna Miller and England soccer player Ashley Cole to the London-based photographic agency Big Pictures.
"The allegations that have been raised are extremely serious and we have launched an immediate investigation," a spokeswoman for the airline said.
"The incident that has been alleged concerns eight customers' flights booked in 2010 and we are in contact with all of those people."
The Guardian newspaper reported that the employee, who looked after high-profile customers for the airline, had now resigned.
It said other celebrities affected included actresses Scarlett Johansson and Gwyneth Paltrow, and singer Robbie Williams.
Miller, star of movies like "Alfie" and "Layer Cake", was awarded 53,000 pounds in damages from Big Pictures in 2008 over claims of paparazzi harassment.
(Reporting by Michael Holden; Editing by Eric Beech)
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