The Italian auto giant scrambled for a succession plan late last week after reports that Marchionne, 66, was gravely ill after suffering serious complications following surgery on his right shoulder last month.
The hospital that treated Marchionne, who died on Wednesday, broke its silence on Thursday.
"Patients' confidence in receiving the best possible treatment and discretion are of utmost importance," the University Hospital Zurich (USZ) said in a statement.
However, the hospital "is being confronted with various rumours in the media about his medical treatment".
"Marchionne was a patient at USZ. Due to serious illness, he had been the recipient of recurring treatment for more than a year," the hospital said. "Although all the options offered by cutting-edge medicine were utilized, Marchionne unfortunately passed away."
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Fiat Chrysler responded by saying: "Due to medical privacy, the company had no knowledge of the facts relating to Marchionne's health.
"The company was made aware that Marchionne had undergone shoulder surgery and released a statement about this," a spokesperson said.
"On Friday July 20th, the company was made aware with no detail by Marchionne's family of the serious deterioration in Marchionne's condition and that as a result he would be unable to return to work. The company promptly took and announced the appropriate action the following day."
Italian news agency ANSA reported late on Thursday that Marchionne's family had confirmed that Fiat Chrysler "was not aware of the state of his health".