The move for the Italy international, who arrives after a loan spell at Cagliari, is reported to be worth around €60million including bonuses, with the Blues seeing off competition from Inter Milan to complete the deal.
Palestra's arrival comes as Chelsea continue a wider rebuild of their defence ahead of Xabi Alonso's first full season in charge.
The club had already announced a costly move back in March, confirming the signing of Portuguese winger Geovany Quenda, 19, from Sporting Lisbon for just over €50million, with Alonso and the club's hierarchy keen to build a squad capable of competing on multiple fronts next season.
Two further defensive signings are also close to completion. The best-known name is Maxence Lacroix, who is expected to join from Crystal Palace for around €55million.
The French centre-back, capped by his country and a member of their World Cup squad, spent four seasons at Wolfsburg before moving to Palace, where he won the FA Cup and the Conference League.
Chelsea are also closing in on a deal for left-back Pep Chavarria, with the two parties in agreement over personal terms. Negotiations with his club, Rayo Vallecano, are ongoing as Chelsea attempt to bring the fee down to below €25million. Bayer Leverkusen had been eyeing Chavarria as a potential replacement for the Spanish left-back Alejandro Grimaldo but have missed out in the race for his signature.
Chelsea are also working on a deal to bring in Granit Xhaka, one of Alonso's top targets for the midfield. The Spanish coach hopes to replicate the success he enjoyed with Xhaka at Bayer Leverkusen now that he is in charge at Stamford Bridge. Xhaka was a key figure as Alonso guided Leverkusen to a Bundesliga and DFB-Pokal double in 2024, before moving on to Sunderland last summer, where he was swiftly named club captain.
Negotiations with the Swiss captain's club, Sunderland, have proven difficult so far.
An initial offer of around €9.3million was rejected, and Chelsea are expected to return with an improved offer in the near future as they continue their pursuit of the 33-year-old.