The unnamed employee stands accused of passing on details of eight high-profile customers of the airline to London-based celebrity photographs agency Big Pictures in 2010.
In a statement, a spokeswoman for Sir Richard Branson's airline said that the allegations were being taken "extremely seriously", with an "immediate investigation" launched.
"The security of customer information is our highest priority and we have robust processes in place to ensure that passenger information is protected," she said.
"The incident that has been alleged concerns eight customers' flights booked in 2010 and we are in contact with all of those people.
"It is too early to draw conclusions on this matter but of course we would deeply regret any concern that these allegations may cause the individuals involved."
The Guardian reported that the employee, who was part of the Virgin Atlantic team looking after high-profile customers, resigned from the airline following the revelations.
The paper said that it had seen email evidence suggesting that Robbie Williams, Scarlett Johansson, Cheryl Cole, Nicole Scherzinger, Gwyneth Paltrow and Tottenham footballer Jermain Defoe may also have been affected by the information leaks.
Legal experts have said that the leak may not constitute a criminal offence, but could be a breach of data protection laws and the celebrity's right to privacy.
Big Pictures, founded by Celebrity Big Brother contestant Darryn Lyons, is well known for its persistent pursuit of celebrities and high-profile figures.
In 2008, actress Miller was awarded £53,000 in damages from Big Pictures over claims that she was harassed by paparazzi.
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