A BAA spokesperson said: "The allegations made were investigated thoroughly and found to be completely unsubstantiated."
Laing O'Rourke refused to comment.
A source close to the case said: "He is a conscientious guy, not somebody to just make something up. But when asked to elaborate he has clammed up and did not give the investigators much to go on."
It is understood that the supervisor made the claims at an interview to select people for redundancy.
The man was to have been made redundant but successfully argued that he ought to be kept on.
Tom Kelly, the GMB's construction organiser, refused to be drawn on the details of the case. He said: "You need to speak to Laing O'Rourke and BAA."
The source said that the future of the man remained unclear.
He said the union now feared that the worker could face disciplinary action if the allegations were deemed malicious.
He said: "It may be difficult for the worker to come back. Laing O'Rourke may want to take action against him for wasting their time, but the workforce may deem him a snitch and seek retribution against him."