He tells Britain's Telegraph magazine, "That did happen a lot (turning down roles). Or rather, it happened a few times that I remember very clearly. Which felt like a lot when I was a little kid. But I think they knew it can be a dangerous business, particularly for a child."
Gyllenhaal instead focused on his studies, and he credits his parents' decision to keep him out of Hollywood with helping him realise his priorities.
He adds, "They really wanted me to have a life, and not just on movie sets. I think they always wanted my sister and me to have a rich life, a life full of things that are full of soul and heart and real connection. That's all they care about.
"My family, as a result, is the most important thing to me. Tomorrow, if Hollywood stopped calling, I'd still have my family."