The centre-back missed the weekend's 3-2 win over Serie A champions Inter with a knee injury that keeps him sidelined for Thursday's (AEST) game.
Juve will have Paulo Dybala available, though, and Pirlo admits he is facing some difficult decisions when it comes to the composition of his forward line.
On Bonucci, the Juventus boss said: "[He] has a knee problem and won't be available tomorrow. We have several solutions for tomorrow."
Dybala also failed to appear in that game, remaining rooted to the bench as Juventus secured a much-needed victory despite losing Rodrigo Bentancur to a 55th-minute red card.
However, the Argentina international is fully fit and could come back into the starting XI in Reggio Emilia.
Pirlo said: "[He] is fine. He played a good game against Sassuolo, but there were no conditions for him to enter on Saturday. He is available and ready to play.
"All the attacking players are doing well, so we will then decide who to start from the beginning."
Juventus' two Serie A meetings with Atalanta this season brought them a home draw and an away defeat.
And Pirlo expects another big test from a team who have continued to grow under Gian Piero Gasperini this term.
He said: "Atalanta forces you to pick up the pace, we know it and we are ready to face it. I don't think it will be a very different match from the other two.
“We earned the final with two hard-fought matches against Inter. We really want to bring this trophy home.”