Add-ons could raise the total cost of the deal to £15 million, it has been reported.
Norgaard is Arsenal's third signing of the summer, following goalkeeper Kepa Arrizabalaga and fellow midfielder Martin Zubimendi.
He could soon be joined by Sporting striker Viktor Gyokeres, who has been heavily linked with the north London club in recent weeks.
Arsenal are set to begin their pre-season campaign later this month against AC Milan.
Norgaard joined Brentford in 2019 and became the club's captain in 2023, under the tutelage of new Tottenham manager Thomas Franck.
The 31-year-old began his professional career with Lyngby in Denmark before moving to Hamburg in 2012.
Norgaard first made a name for himself at Brondby, making 147 appearances between 2013 and 2018.
He signed for Fiorentina in 2018 but struggled to establish himself before moving to the Premier League.