The 25-year-old left the Danish camp after suffering a thigh injury during Thursday’s 3-2 defeat to Norway, with head coach Brian Riemer confirming the news.
“He is out of this Nations League, yes,” Riemer told TV 2.
Grønbæk was introduced in the 82nd minute against Norway and completed the match, but was later seen with his thigh heavily bandaged.
The attacking midfielder has started all five of Hamburg’s competitive matches this season, scoring twice and providing two assists since joining the club.
The injury comes as Hamburg continue to build their squad following a return to the Bundesliga.
According to Bild, sporting director Claus Costa has been given greater freedom to make significant investments when he believes they make sporting sense.
Hamburg spent €9 million to sign Fábio Vieira from Arsenal and €8 million on Zakaria El Ouahdi from Genk during the summer.
The club’s improved financial position has helped create the room for such deals, with Hamburg reporting record revenue of more than €182 million last season and a profit exceeding €8 million.
“We now earn almost three out of every four euros outside of media rights and transfers,” board member Eric Huwer said.
“Financial stability is the condition for sporting freedom of action, even in more difficult phases.”
Hamburg can therefore reinvest additional income from sporting success or future transfer sales into the squad, although Grønbæk’s immediate absence is now another issue for the club to manage.