Schweinsteiger suffered damage to his medial collateral knee ligament during training ahead of Saturday's international friendly against England in Berlin.
The 31-year-old midfielder has struggled for form and fitness at Manchester United this season, having only recently returned from a two-month lay-off due to an injury to the same knee.
But Hitzfeld, who coached Schweinsteiger over two separate spells with Bayern, still believes his absence would be keenly felt by Joachim Low's squad.
"It would be sad for German football if he is were out of the Euros because this tournament is still a great goal for him," Hitzfeld told Abendzeitung.
"That would be a big loss for the national team. Also, a human loss – for Bastian is the captain and holds the team together. He is still an absolute role model.
"If he should be fit for the European Championship, it will be an important component for Joachim Low, if Bastian can bring consistent performances.
"If he gets match practice and, therefore physically fit, he is still a highly intelligent player."
World Cup-winner Schweinsteiger has 114 caps for his country, scoring 23 times.