According to _Sky Sport_, Chelsea have made contact with the entourage of the former Cagliari defender, though the approach is currently limited to a request for information rather than the opening of formal negotiations.
Chelsea's interest comes shortly after the west London club sold Marc Cucurella to Real Madrid, leaving them in need of a long-term option on the left side of their defence.
Palestra's versatility, with the ability to operate on both flanks, makes him an appealing fit for the role Chelsea are looking to fill.
Despite the growing interest from England, reports suggest the priority for both the 21-year-old and his agent Alessandro Lucci is to remain in Italy, in a league he already knows well.
That preference puts Inter Milan in pole position, though the Serie A champions have so far been unable to reach an agreement with Atalanta on a fee for the player.
Atalanta are demanding at least €50 million for Palestra, a figure that may prove difficult for Inter to meet given continued interest from clubs abroad.
Manchester City have already reached an agreement with Palestra on personal terms, according to Italian reports, positioning the Premier League champions as a serious threat to hijack any move to Inter Milan.
Palestra has built his reputation through a breakout loan season at Cagliari, where he made 37 Serie A appearances, contributed one goal and four assists, and established himself as one of the most heavily scouted young full-backs in European football.
Across his career to date, the 21-year-old has made 41 appearances and scored three goals for Atalanta's Under-23 side, alongside 16 senior appearances for Atalanta's first team and 37 appearances for Cagliari during his loan.
He has also represented Italy at every age group from Under-18 through to full senior level, winning two caps for the senior national team alongside extensive underage international experience.
Palestra's contract with Atalanta runs until June 2029, giving the Bergamo club significant control over the timing and terms of any departure.
His honours include a UEFA Europa League title, won during his time in Atalanta's youth and senior setup, adding further weight to his growing reputation across the continent.
With Chelsea's interest still at an early stage and Inter Milan yet to find a breakthrough on price, Manchester City's head start on personal terms may prove decisive — unless Palestra's stated wish to stay in Italy ultimately tips the balance in Inter's favour.