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Marcos Rojo is fit, but Chris Smalling (leg), Phil Jones (hamstring) and Jonny Evans (ankle) are injured and Tyler Blackett is suspended.
Van Gaal said: "All three of my right-sided centre-halves are out, so I'll have to go to the youth system."
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Potential candidates to play include Tom Thorpe, 21, and Paddy McNair, 19.
Neither under-21 captain Thorpe nor McNair, who joined United as a midfielder three years ago, have played for the first team.
Smalling limped out of training on Wednesday and, with Luke Shaw expected to make his debut at left-back following his £28m summer arrival from Southampton, Rojo is likely to move to centre-half against the Hammers.
"I won't take risks with players - they have to be fit," said Van Gaal, who is without nine players in total through injury.
Midfielders Daley Blind or Darren Fletcher could be employed as centre-backs by United, who are 12th in the table with only one win from their first five games.
But Van Gaal denied it would be a risk to select the younger players.
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"They have hunger to play the game and the Manchester United youth education system has a certain level that can provide that," said the Dutchman.
Van Gaal admitted there was a "big difference" between youth football and Premier League, but added: "In one game you can show yourself."
Asked if there was a sense of panic after United let slip a 3-1 lead in losing 5-3 at Leicester City last weekend, he said: "No. It's happened - it's about overcoming things.
"It's not a regular occurrence. When you see our figures in the Premier League - with the exception of the Leicester match - then we are defending well.
"I don't think my players are robots - they are human beings and human beings make errors."
Manager Louis van Gaal said Manchester United have only one senior centre-back available for Saturday's Premier League game against West Ham.