While cancer is not contagious, there have been rare cases of cancer transferring from one person to another during organ transplants. Physicians carefully screen donors to reduce this risk; however, some cases have occurred when an organ transplanted from someone with cancer causes the same type of cancer in the recipient. Keep in mind, however, that a person receiving an organ takes medication that weakens his or her immune system – making that patient’s body unable to fight off cancer cells as it normally would.