Shaw suffered a head injury during the 2-1 win at Crystal Palace last weekend, while Rooney was forced off in the same game with a dead leg.
And speaking at his pre-match news conference on Friday afternoon, Van Gaal confirmed that the duo would miss out.
However, there was some good news for United after Van Gaal confirmed that striker Robin van Persie, winger Angel di Maria and defender Marcos Rojo are all available for selection after various spells on the sidelines.
Van Gaal told a news conference at Carrington on Friday: “The situation is not positive for us, they cannot play, neither of them.
“Yes it is a short term thing, we have only one match to play, so I think it will be difficult for them to play again this season.”
Arsenal head to Old Trafford looking to return to winning ways after their surprise 1-0 defeat by Swansea City at Emirates Stadium on Monday night.
The north London side are currently third in the table heading into their final three Premier League games of the season.
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