As the row continued to fester a week after the comments were made, Chiellini said the position needed somebody with "decorum and personality" and that Tavecchio's rival Demetrio Albertini, a former Italy midfielder, was the only suitable candidate.
Tavecchio gave a radio interview later on Sunday in which he said he suffered more than Lee Harvey Oswald, who assassinated American President John F. Kennedy in 1963.
“John Kennedy’s assassin didn’t have to suffer what I have suffered over the last few days,” Tavecchio told RAI radio.
“And to think that these last few days mark 30 years since I started my work in Africa, where I have helped to build and open a hospital.”
The election to replace Giancarlo Abete, who quit following Italy's group-stage exit at the World Cup, is due to take place on Aug. 11.
"I read his (Tavecchio's) words the day after and they left me disconcerted," Chiellini told 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," said the 29-year-old.
"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."
Chiellini was already in the spotlight after being bitten by Uruguay forward Luis Suarez during a World Cup group match in June.
"I have nothing personal against Tavecchio but at the moment, the only adequate person for the role is Albertini," said the 29-year-old.
FIGC vice-president Tavecchio has received a wave of criticism for his comments, which came at the summer assembly of Italy's amateur leagues (LND), and FIFA has asked for an investigation.
He made the comment as he suggested that Italy should replicate England's stringent requirements for non-EU players.
"In England, they identify the players coming in and, if they are professional, they are allowed to play," he said.
"Here instead we get 'Opti Poba', who previously ate bananas and then suddenly becomes a first-team player with Lazio."
Tavecchio, who initially had the support of all Serie A clubs except Juventus and AS Roma, has seen his backing drop since the incident, although many clubs, including AC Milan, still back him.
Roma goalkeeper Morgan De Sanctis also criticised Tavecchio.
“He’s not suitable, if the FIGC want to change, that they cannot do it with a figure such as him,” he told reporters. “We need something different.”
(Reporting by Brian Homewood, editing by Ed Osmond)
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