With an online report saying the 74-year-old ex-boxer was on life support, Gunnell said that every time Ali was in the hospital it triggered a "media frenzy," the Courier-Journal reported.
Gunnell had disclosed on Thursday that Ali was in a hospital, which has not been identified, in fair condition. He did not return repeated requests for comment.
Ali has suffered from Parkinson's disease for more than three decades and has kept a low profile in recent years.
The New York Post and the International Business Times cited a report on the Radar Online website that said Ali had been placed on life support.
Radar Online quoted "an insider" as saying, "Doctors are telling the family that it likely won’t be long until he passes away."
Britain's Mirror reported that four of his nine children had rushed to the former champion's side.
The Associated Press cited two people familiar with his condition as saying that Ali was facing more serious health problems than in his previous hospital stays because of his Parkinson's disease.
Ali's last public appearance was in April at the "Celebrity Fight Night" gala in Arizona, a charity that benefits the Muhammad Ali Parkinson Center.
At the height of his career, Ali was known for his dancing feet and quick fists and his ability, as he put it, to float like a butterfly and sting like a bee.
Ali, who lives in Phoenix, was admitted to hospital in December 2014 with what was believed to be a mild case of pneumonia.
Doctors determined he had suffered a severe urinary tract infection, for which he received further treatment in January 2015.
Nicknamed "The Greatest," he retired from boxing in 1981 with a 56-5 record. Ali's diagnosis of Parkinson's came about three years after he retired.
Ali, born in Louisville, Kentucky, as Cassius Marcellus Clay Jr., changed his name in 1964 after his conversion to Islam.
(Reporting by Ian Simpson in Washington; Editing by Matthew Lewis)
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