Burns began playing for Celtic in 1974 and also won eight Scotland caps in a distinguished career.
He went on to manage the Parkhead club and steered them to a Scottish Cup win over Airdrie in 1995.
He has also managed Kilmarnock and Reading.
In March 2006, Burns underwent treatment for skin cancer, but it returned two years later.
"Clearly, all right-minded people will understand that the club is not in a position to provide updates in terms of Tommy's condition," the Celtic statement added.
"And once again, we would ask that the privacy of Tommy and his family is fully respected at this time."