2018 FIFA World Cup-winning France coach Didier Deschamps won the award for Best Coach at the governing body's ceremony in London, while Croatia captain Luka Modric won Best Men's Player of the Year, breaking the 10-year Messi-Ronaldo duopoly.
National team captains and coaches, and a media representative, are allocated votes for the awards - but Chiellini left Allegri out of his top three entirely.
They won a Serie A and Coppa Italia double together last season, but the towering defender voted for, in order, Deschamps, former Real Madrid coach Zinedine Zidane, and England coach Gareth Southgate.
He did at least vote for his club teammate Cristiano Ronaldo first in the Best Player category, followed by Luka Modric and Egyptian Mohamed Salah.
Italy coach Roberto Mancini voted for Deschamps, Croatia's Zlatko Dalic, and Allegri, while Italy's media representative Fabio Licari voted for Deschamps, Zidane and Allegri.
