Palestra, a Milan native who spent time in Inter Milan's academy as a boy, had the chance to join the newly-crowned Italian champions this summer instead of moving to Stamford Bridge in a £47m deal from Atalanta.
Instead, it was a call from his new head coach that swayed him.
"He likes me," Palestra says of the conversation with Alonso. "He likes my flexibility because I can play on the right, left, lower, higher. There were some clubs that wanted me but Chelsea, from the first moment they wanted me, I agreed, because they're one of the biggest clubs in the world. They play in the Premier League, which is the best in the world. I'm excited."
The 21-year-old arrives off the back of a stellar season on loan at Cagliari, where he was named Serie A's Defender of the Year and won the division's Frecciarossa Speed Award, before capping it off with his senior Italy debut. He is well aware of the club's Italian pedigree, name-checking Gianfranco Zola unprompted during the conversation.
"I didn't call him, but I know Zola played at Cagliari and at Chelsea. He's a legend. Maybe I can do what he did at Chelsea," he says, smiling, before adding when asked if he wants to become a Chelsea legend himself: "Maybe, maybe, I hope. I know some Italian players came to Chelsea.
"I spoke with Davide Zappacosta after I joined Chelsea. I played with him at Atalanta. He's a top guy. I have a good relationship with Zappa. He always talked good things from Chelsea. I know a lot of Italian players played here. I want to be the next top one."
Zola, Gianluca Vialli, Roberto Di Matteo, Jorginho and Carlo Cudicini have all left their mark at Stamford Bridge before him.
Chelsea captain Reece James was quick to make him feel welcome, too.
"He wrote to me when I joined Chelsea, a beautiful message, how happy he is that I joined, and I said hi to him when he came to Cobham one day after the World Cup," Palestra says. "I know he is a good guy, a good captain, so I can't wait to play with him. I'm very happy."
Palestra's rise has been rapid given a slow start: as a teenager he says he was the slowest player in his team before a growth spurt transformed him into one of Serie A's quickest.
"When I was young, like 15, I remember I was the slowest one in the team, but after maybe one or two years, I grew physically and I changed position. Before, I played in midfield, but I changed position to left back or at the back, and I understood that I am quick. When I was young, I'm the slowest. It's crazy."
Whether he is now the fastest at Chelsea remains to be settled.
"I don't know, I need to race all of my team-mates!" he laughs, before reflecting on his rapid rise. "At the start of last season, I couldn't imagine I would win Serie A Defender of the Year. It's a big honour to be the best defender but I know it is only the start. I need to improve every day."
Versatile enough to play at full-back or wing-back on either flank, Palestra has already impressed in pre-season.
With Italy having missed out on this summer's World Cup, he sees his move to England as part of the solution.
"Italy is not in the best moment of its life and it's a good thing I came to the Premier League because it's the best in the world. I can prove myself more. We can help Italy get to the World Cup, and to win something."
Asked whether he could become the Premier League's best defender, he stays grounded: "I hope! Maybe it's difficult, but I said the same at the start of the last season. I will try to do my best. It's crazy because when I was young, always my aim was to play at this level. It's a dream. The first thing I want is to win for Chelsea, always the team. If after the season, I also win the Defender of the Year, then I am happy, but I prefer to win something for Chelsea and nothing for me."