Tavecchio caused uproar last Saturday when he referred to African footballers as "banana eaters" in a speech calling for work permits to be introduced into Italy's professional football divisions.
The 71-year-old has since apologised for the remarks but the fall-out has continued, with FIFA asking the FIGC to investigate Tavecchio's words.
An election to replace Giancarlo Abete, who left his post following Italy's group stage exit at the World Cup, is to be held on August 11.
Tavecchio previously had the backing of all Serie A clubs bar Juventus and Roma prior to the incident, but has since reportedly seen support for his election decrease.
And Chiellini feels that Tavecchio's opponent Demetrio Albertini is the only option for Italian football's governing body in the wake of the comments.
"I read his [Tavecchio's] words the day after and they left me disconcerted," Chiellini said at a Juventus news conference on Sunday.
"I asked... how Italian football could be heading towards this election and if there was any solution. I have nothing personal against Tavecchio but at the moment, the only adequate person for the role is Albertini.
"We mustn't let Italian football embarrass itself again. The national team is bigger than Tavecchio but we must avoid having a certain type of figure for these roles.
"We need a person with a certain decorum and personality to perform certain roles."
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