The 21-year-old midfielder was a standout performer for Denmark at Euro 2020, scoring two goals in five appearances, including a stunning free-kick against England in the semi-finals.
According to Sportmediaset, the four English top-flight clubs as well as Barcelona and Juventus have all expressed their interest in Damsgaard.
Sampdoria are aware of interest in the midfielder and are willing to accept any offers in the region on £30m.
The Italian outfit only signed Damsgaard from Danish side FC Nordsjaelland in July 2020 for around £5.7m, but his sale this summer could help enhance their transfer budget.
Damsgaard, who has three years remaining on his contract, made 35 Serie A appearances last season, scoring two goals and providing four assists.