It is the second time in three seasons Juve have made the quarter-finals, but they have not progressed beyond that stage since 2002-03.
They join Paris Saint-Germain, Barcelona, Atletico Madrid, Monaco, Porto, Real Madrid and Bayern Munich in the last eight, and it is the Bundesliga champions who Chiellini fears most.
"If I can, I would avoid Bayern," the 30-year-old centre-back told reporters. "But we will still watch the draw with great emotion."
Bianconeri team-mate Leonardo Bonucci, on the other hand, is adamant Massimiliano Allegri's men would be happy facing any one of those seven sides.
"There are still lots of matches to play and this is where the Champions League starts to get interesting," the defender said.
"We're through to the quarters and we'll head into them with a bounce in our step, fearing no one.
"Whoever we get drawn against is going to be tough but we know we'll be a hard team to beat as well."