The 74-year-old collapsed twice at Littlehey Prison in Cambridgeshire, where he was serving an eight-year jail sentence for historic sex offences.
A spokesman for the Ministry of Justice confirmed Clifford died in hospital on December 10.
"As with all deaths in custody, there will be an investigation by the independent Prisons and Probation Ombudsman," the spokesman said.
"Our condolences are with Mr Clifford's family at this difficult time."
Clifford's daughter Louise told the Mail on Sunday that her father first collapsed in his cell while he was trying to clean it on Thursday.
"It was just too much," she said.
"Next day he collapsed again and was unconscious for several minutes - though he doesn't know how long.
"He was seen by a nurse, who insisted he must be transferred to a local hospital.
"That's where he had his cardiac arrest, later on Friday. He is now in the critical care unit. He's in a bad way."
Clifford was jailed in May 2014 after he was convicted of eight counts of indecent assault against four women between 1977 and 1984.
Passing sentence at Southwark Crown Court, Judge Anthony Leonard told Clifford he had led a "double existence", living an outwardly respectable existence while grooming young girls for abuse.
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