Spinks, 66, also is the father of former world champion boxer Cory Spinks and the brother of former world champion Michael Spinks. His family did not disclose the nature of his illness. He is being treated in the Las Vegas area.
"Leon is currently hospitalized and in the care of an exceptional team of doctors," the Spinks family said in a statement Thursday. "We kindly request that privacy be given to his family at this time."
Spinks won a gold medal in the 1976 Olympics and was inducted into the Nevada Boxing Hall of Fame in 2017.
"Neon Leon" scored one of the best upsets in boxing history when he beat Ali by split decision in 1978 in Las Vegas. The ex-Marine claimed the WBA, WBC and lineal heavyweight championships by beating Ali in the 15-round bout.
Spinks' unlikely triumph snapped a 14-fight win streak for Ali and was Ali's first in-ring championship loss.
Ali snatched the title back by beating Spinks in a rematch seven months later in New Orleans.
"Dear Friends: I know it's been a while," Spinks' wife, Blenda Glur Spinks, wrote Monday on Facebook. "However, you know how I believe in the power of prayer.
"It's been a tough year for us. Leon has endured a lot of medical problems. I'm reaching out to ask you to kindly pray for my beautiful husband Leon. So that he may overcome the obstacles that have crossed his path. We love you."
Spinks had a 26-17-3 record with 14 knockouts. His last professional fight was a loss to Fred Houpe in 1995.