The 22-year-old was one of the standout performers at the 2020 European Championships for Denmark, during which he scored two goals and registered one assist to help his side reach the semi-finals.
Damsgaard netted the opening goal from an excellent free kick in the 2-1 extra-time defeat to England at Wembley Stadium, but the midfielder has struggled to replicate that kind of form for his club.
A persistent thigh injury ruled Damsgaard out of 27 Serie A games for Sampdoria last season, with a lack of form meaning he failed to register either a goal or an assist in his 11 league appearances.
That has not put Thomas Frank's Brentford side off this summer though, with journalist Gianluca di Marzio reporting that the Bees have submitted an offer of €11m (£9.2m) plus an additional €7m (£5.9m) in bonuses for the Dane.
The report suggests that the Italian outfit will demand more than this proposed offer, but believe that an agreement can be reached.
Should Damsgaard make the move to West London, he would join a strong Danish contingent at the Premier League club, with the likes of head coach Frank, as well as Luka Racic, Mads Bech Sorensen, Mads Roerslev, Mathias Jensen and Christian Norgaard also coming from the Scandinavian country.
Christian Eriksen is one member of the Danish contingent to have left Brentford this summer though, having joined Manchester United upon the expiration of his contract.
However, the Bees have strengthened their squad with the arrivals of Keane Lewis-Potter, Aaron Hickey, Ben Mee and Thomas Strakosha during the transfer window so far.