David Aldridge of Los Angeles had a kidney transplant in 2006, but he will soon need another. Like many people living with HIV, he suffers from kidney damage, either from the virus or from the life-saving medications that keep it at bay. Until recently, such patients did not receive transplants at all because doctors worried that their health was too compromised. Now they can get transplants, but organ-donor waiting lists are long. And for Mr. Aldridge, 45, and other HIV patients, a potential source of kidneys and livers is off limits, because it is illegal to transplant organs from donors who test positive for the virus—even to others who test positive. But federal health officials and other experts are calling for repeal of the provision that bans such transplants, a 23-year-old amendment to the National Organ Transplant Act.