Trump said in an interview taped before the Super Bowl that football is "really tough."
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He says equipment, including helmets, has improved "but it hasn't solved the problem."
Trump thinks the NFL "is a great product." But as for Barron playing, the president calls it a "very tough question."
"If he wanted to? Yes. Would I steer him that way? No, I wouldn't."
The president says many people, "including me, thought soccer would probably never make it in this country, but it really is moving forward rapidly."
Trump has, in the past, bemoaned that football games have become less violent. The NFL and college football have increased penalties and enforcement for illegal hits to the head and for hitting defenseless players.
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"They're ruining the game," he said during a rally in Alabama in September 2017.
He said players were being thrown out for aggressive tackles, and it's "not the same game."
President Barack Obama, the father of two daughters, said in a 2013 interview with the New Republic that he would "have to think long and hard" before letting a son, if he had one, play football because of the risk of head injuries.
Obama also said football may need to change to prevent injuries.