"It is very sad news for us and my brothers and sisters and all the members of my family."
Mr Elliott was to turn 80 in 10 days.
Melbourne-born Mr Elliott is understood to have been hospitalised after a recent fall.
In a statement, Carlton noted Mr Elliott's passing with sadness.
"He led the Blues through a record 20-year presidency, with the larger-than-life figure overseeing a significant period of on-and-off field success throughout his tenure," the club said.
Tom acknowledged his father's commitment to the Blues.
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"Apart from his family, Carlton was the great love of his life."
Federal Treasurer Josh Frydenberg was among the first public figures to pay tribute to Mr Elliott.
"He was a proud Victorian and a larger-than-life figure," he tweeted.
Mr Elliott stayed on the party rather than parliamentary side of politics, serving as federal Liberal treasurer and president from the 1980s.
Once touted as a potential prime minister, Mr Elliott chose business over parliament.
His business life included the acquisition of multi-million dollar companies including in the brewing industry.
After a turn in fortune, Mr Elliott declared himself bankrupt in 2005. After the bankruptcy was lifted, he maintained a public profile, including with speaking engagements.
"John Elliott was a huge character," Melbourne-based comedian Dave Hughes said in a tweet on Thursday night.
Mr Elliott was a father to four: Tom, Caroline, Edward and Alexandra.
He was also a grandfather to six.