"You can never say [how long he could be out for]. He has a knee problem so we must wait and see."
It comes at a crucial point in the season with Van Gaal's side set to face Shrewsbury in the FA Cup on Monday once they return from Denmark.
It will also be a worry for England boss Roy Hodgson ahead of Euro 2016.
England have friendlies against Germany and Holland next month, and there are bound to be concerns that Rooney could go into yet another major championship not in peak condition.
But Rooney might miss as many as 11 matches for United, including games against Arsenal, Manchester City, Everton and Tottenham.
He could miss as many as eight Premier League games for Van Gaal's side.
Matteo Darmian also misses out for United on Thursday. Van Gaal said: "Darmian has a dislocated shoulder. It's not the heaviest type so we are lucky, but he's injured and out."
Van Gaal also confirmed that Cameron Borthwick-Jackson has not travelled to Denmark due to illness.
MATCHES ROONEY COULD MISS
Feb 18 - Europa League - Midtjylland (A)
Feb 22 - FA Cup - Shrewsbury (A)
Feb 25 - Europa League - Midtjylland (H)
Feb 28 - Premier League - Arsenal (H)
Mar 2 - Premier League - Watford (H)
Mar 6 - Premier League - West Brom (A)
Mar 12 - Premier League - Crystal Palace (H)
Mar 20 - Premier League - Manchester City (A)
Apr 2 - Premier League - Everton (H)
Apr 9 - Premier League - Tottenham (A)
Apr 16 - Premier League - Aston Villa (H)
ENGLAND
Mar 26 - Friendly - Germany (A)
Mar 29 - Friendly - Holland (H)