Damsgaard has caught the eye so far at Euro 2020, with vital goals for Denmark in their push to reach the last four of the competition.
Th 21-year-old has continued his impressive domestic form onto the international stage in recent weeks, with La Blaugrana rumoured to be monitoring him ahead of a possible summer bid.
According to reports from Mundo Deportivo, the Catalan giants are keen to solidify their interest in the coming weeks to ward off any potential rivals for him.
Despite the rumoured spike in interest in Damsgaard, Sampdoria are under no pressure to sell their key man, and the Italian club could demand in the region of €35m for him this summer.
Barcelona are likely to make one major splash in the summer market, following a string of free transfer moves, and a move for Damsgaard could fit their current model.
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