Sport says Barcelona are following his performances at Euro 2020.
He scored a stunning goal against Russia in the final group game as Denmark won 4-1 to progress as runners-up in Group B, behind Belgium. Damsgaard received the ball on the edge of the box, cut back to position his body and sent the ball flying into the back of the net.
Sampdoria's president, Massimo Ferrero, commented on Il Secolo XIX: "Did you see Damsgaard's goal? I'll tell you, his price right now is at least 30 million euros. But if he scores again, it will reach 40 million."
The Serie A side are preparing for an auction for the young star, who at 20 has also been linked with clubs in Italy and in England, in addition to Barça.