The 25-year-old is running down the months on his Cherries contract, which is due to expire in the summer of 2024, and Andoni Iraola's side do not possess the option to extend his stay.
Kelly has struggled to stay fit throughout the 2023-24 campaign, making seven appearances for Bournemouth in all competitions, including five starts in the Premier League.
However, the former Bristol City product wore the captain's armband in last weekend's meeting with Burnley, where the Cherries claimed their first top-flight victory of the new term in a 2-1 win.
Bournemouth's success saw Iraola's men shoot out of the relegation zone into 17th place, but they remain very much in the demotion mix and would be resigned to losing Kelly on a free transfer should they go down to the Championship.
The Englishman - who represented his nation at Under-21 level - was briefly linked with a move to Tottenham Hotspur over the summer, before the Lilywhites opted to sign Micky van de Ven instead.
Interest in Kelly's services is expected to grow should it become clear that he will not be extending his Bournemouth deal, though, and Tottenham could reignite their interest as they search for more defensive reinforcements in 2024.
However, journalist Fabrizio Romano reports that the Premier League leaders could face stiff competition from AC Milan, who have identified Kelly as an ideal acquisition for their youthful recruitment drive.
Romano states that the Rossoneri are working to bring Kelly to San Siro as early as January, which could be Bournemouth's last chance to recoup a fee for the defender before he enters pre-contract talks with foreign clubs.
Milan have been left short of central defensive options for the winter period, as Mattia Caldara and Marco Pellegrino are both expected to be missing for a few weeks due to ankle injuries, while Pierre Kalulu sustained a thigh issue against Napoli and faces around four months on the sidelines.
As a result, Stefano Pioli is working with Fikayo Tomori, Malick Thiaw and Simon Kjaer as his fit centre-halfs, but the latter will turn 35 next year and is also due to become a free agent at the end of the season.
Thiaw has also been linked with a move to Real Madrid next year, but Milan are unlikely to consider selling the Germany international, who is also thought to be happy at San Siro.
Alongside Milan, however, Juventus are also rumoured to be keeping tabs on Kelly's situation, and the Bianconeri have also been informed of the conditions of any potential deal.
Bournemouth forked out €14.8m (£12.9m) to bring Kelly to the Vitality Stadium from Bristol City in 2019, and the 25-year-old boasts two goals and seven assists in 129 matches for the club.
Kelly's favoured position is in the heart of the backline, but the former England youth international is also comfortable as a left-back.