The 20-year-old joined Sampdoria last summer from Danish top flight club Nordsjaelland on a four-year contract.
Damsgaard plays a key role at the Serie A outfit, playing a part in 23 Serie A outings so far this season, with his performances drawing admiring glances from Premier League sides Spurs and Leicester, along with Serie A giants Juventus, Inter and Napoli.
However, despite the interest from several European heavyweights, the Dane has revealed he is open to staying at the Luigi Ferraris for a few more years.
Asked whether he is open to staying at Sampdoria for a few more years, Damsgaard told Italian daily Il Secolo XIX: “Of course, why not?
“Then of course, in life you never know and in football everything depends on performance, the good but also the bad.”
The winger admitted he is proud that the Sampdoria faithful love him as a player, but chose to remain humble and stressed he does not believe he has done enough to deserve it.
“I don’t read news like this [about the Sampdoria faithful wanting me to stay] but if that’s the case I’m very proud of it.
“With only two goals scored, I don’t think I’ve done so much to deserve it.”
Damsgaard has predominantly played a winger this season while he is also comfortable in the central midfield role or as an attacking midfielder.