Burns, 51, was first diagnosed as suffering from skin cancer in 2006 but had responded well to treatment.
However, a spokesperson for the club said: "Celtic FC today confirmed that first-team coach Tommy Burns is to receive continued cancer treatment.
"Tommy intends to continue in his role as first-team coach and Head of Youth Academy during this period of treatment.
"Clearly, Celtic Football Club requests that the privacy of Tommy and his family is respected at this time."