The 31-year-old suffered a knee injury on international duty last week during training for Germany's 3-2 friendly defeat by England.
"Schweinsteiger is a tragic injury because the expectation is not so good for him in respect of the European Championship but you never know," Van Gaal told a news conference ahead of United's Premier League game at home to Everton on Sunday.
"He has to work hard for that because he wants to play and the Germany national manager wants him to play, so I hope that he can reach that, but it shall be difficult," the Dutchman added.
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Schweinsteiger, who helped Germany win the 2014 World Cup, returned to United's first team three weeks ago after two months out with another knee problem in an injury-plagued first season at Old Trafford.
The European Championship starts on June 10 in France.
(Reporting by Ed Osmond, editing by Pritha Sarkar)
