Bove progressed through the youth ranks of his boyhood club, before being awarded his senior debut by Jose Mourinho in May 2021.
After going on to make 13 appearances across the 2021-22 campaign, he saw his playing time increase during the course of last season.
Bove featured in 33 competitive games throughout the 2022-23 season, including 22 appearances in the Italian top flight.
The 21-year-old has already appeared in 32 matches during the current campaign, chipping in with one goal and two assists.
According to TuttoMercatoWeb, Bove's performances have caught the attention of Premier League duo Bournemouth and West Ham.
The report suggests the two English clubs have been monitoring Bove's progress, as they weigh up whether to pursue a deal in the summer transfer window.
The Cherries are seemingly considering the prospect of adding to their current central midfield options, which include Lewis Cook, Alex Scott, Philip Billing and Tyler Adams.
Meanwhile, West Ham may need to strengthen in the middle of the park if they decide against pursuing a permanent deal for Manchester City loanee Kalvin Phillips.
While Bove may be attracting interest from the Premier League, Roma are under no pressure to part ways with their academy product.
The youngster put pen to paper on a new deal in December to extend his stay with the Giallorossi until the summer of 2028.
With Bove under contract for the next four years, it would reportedly take a significant offer for Roma to consider selling the Italy Under-21 international.
Bove is currently focused on helping Roma on the pitch, with the Italian club currently in sixth place in Serie A and in the last 16 of the Europa League.