The Bianconeri's previous attempt to claim all three trophies in the same season fell at the final hurdle in 2015, with Pirlo part of the side that suffered a 3-1 defeat against Barcelona in the Champions League showpiece in Berlin.
Although the experienced midfielder now plies his trade at New York City, many of his old team-mates remain in the Juve side and he is confident they can improve on their last showing at this stage.
"Every now and then I think back to that night two years ago in Berlin, because when you lose a final, it comes back into your thoughts and you'd love to play it again," he wrote in La Stampa.
"Unfortunately for me, I can't do that, but I am happy for them.
"And I am happy for Gianluigi Buffon. He's been waiting a lifetime to lift this cup. I hope he wins it, after coming so close at other times. I say this as a friend and a former team-mate, someone who knows him well. He deserves it, it's the only thing he's missing.
"I read that he could win the Ballon d'Or; he hasn't surprised me, not even in this extraordinary season, because when he truly believes in something and has that objective in his head, he'll do anything to reach it.
"Similarly, I was not surprised by Juve, even if it's not easy to win everything, as I knew this season they were targeting the Champions League. I expected nothing other than this.
"I remember in December, when I went back to Turin and had dinner with Gigi, chatting to other former team-mates too, they were convinced they had a great squad and could go all the way.
"There was that confidence and hunger, which I too have known. A final is not yet the end and Real are a great team; there's reason to fear, but that's true of all opponents at this level.
"If Juve face this test the way they know how, they've got a big chance of winning. At the end of the day, this season they never got it wrong in the games that matter.
"I know that in July you will come to play around these parts and I'll visit. I'd happily say hello to the cup."