Damsgaard joined Italian top flight club Sampdoria from Danish outfit Nordsjaelland for a fee in the region of €6.5m during the last summer transfer window.
The winger has gone on to become a regular starter for the Serie A club and has contributed to six goals from 24 league appearances so far this campaign.
Damsgaard’s performances during his debut season for Sampdoria have seen him attract the interest of a host of clubs, including Tottenham and Leeds in the Premier League.
According to Italian outlet Calciomercato.com, the Serie A outfit value the Denmark international at €20m as the player continue to attract more interest.
Apart from Tottenham and Leeds, Damsgard has also been associated with a move to West Ham and Leicester City recently.
Serie A giants Inter, Juventus and Napoli are also claimed to be keen to acquire his services, as a transfer scramble for him in the summer threatens to break out.
With Sampdoria slapping a €20m price tag on him, it remains to be seen who Damsgaard will be plying his trade for by the end of the forthcoming transfer window.