The Danish international, who impressed at Euro 2020 last summer, can play on the left-wing or in the no.10 role.
According to the Daily Mail, the 22-year-old is seen as the ideal replacement for his countryman Christian Eriksen, who left on a free transfer to Manchester United.
Damsgaard struggled with injuries after his Euros heroics for Denmark, only managing 12 appearances last term.
But he is now fully fit and Brentford believe he has what it takes to succeed in the Premier League.
Damsgaard will join a significant Danish contingent already at Brentford, including manager Thomas Frank.
