According to a report from AS, the 20-year-old midfielder is being heavily scouted by Leicester and Spurs, seeing him as a potential transfer option for the summer.
Damsgaard is one of Sampdoria's most highly rated players, and the Italian club are set to demand a premium for him in the summer, especially after paying €7 million last summer to sign him from Nordsjaelland.
Sampdoria reportedly want three times the price that they bought him for, and if Damsgaard has a good end to the season, they might even demand more.
The 20-year-old has bagged two goals and four assists this season, playing as either a left-winger or else as a central attacking midfielder.
Leicester are in need of some more depth, especially since they will most likely not make the loan move for Cengiz Under from Roma permanent due to him failing to settle down at the club.
On the other hand, Spurs are keen on signing a new winger, and with Erik Lamela set to leave the club for Italy this summer, Damsgaard looks like a great replacement.
Damsgaard is a very talented player on the ball, and with the right coaching and experience, he will surely get even better given his age.
Even if the likes of Leicester and Spurs are forced to pay a significant fee for the Danish winger, he will have lots of time in his career to justify the amount paid for him.