The Germany international has become one of the most sought-after young forwards in European football, with his performances in Belgium attracting attention from a growing list of potential suitors.
According to reports in Italy, Chelsea and Manchester City have sent scouts to watch Tresoldi on several occasions, while Arsenal, Tottenham and Newcastle have also been represented at Club Brugge matches.
The interest extends beyond England. Borussia Dortmund and Bayer Leverkusen are among the German clubs linked with the forward, while Atletico Madrid and Juventus are also monitoring his situation.
AC Milan are particularly keen, with Tresoldi understood to have grown up supporting the Italian club. However, that personal connection may not be enough to give the Rossoneri an advantage if Premier League clubs make substantial offers.
Tresoldi was born in Cagliari but developed in Germany before joining Club Brugge. His performances this season have strengthened his reputation, with 14 goals in 16 appearances across competitions.
His scoring record has made him an attractive proposition for clubs looking to invest in a forward with significant potential and an ability to contribute immediately.
Club Brugge are in no hurry to sell, however, and would prefer to keep the striker at least until the end of the season.
The Belgian club's valuation has reportedly risen considerably. Figures of €25 million to €30 million circulated previously, but the latest reports suggest they now want around €50 million, with the possibility that the asking price could increase further.
That figure would make a January deal difficult, particularly with Brugge determined to retain one of their most important attacking players.
For Milan, the challenge will be balancing their interest with the financial strength of their rivals. Chelsea and Manchester City have the resources to compete for a player who is attracting attention across Europe.
With so many clubs monitoring his progress, Tresoldi's future is likely to become one of the more closely followed transfer stories ahead of the next summer window.