Russell, whose 18-year-old son suffers from bipolar disorder, told Katie Couric that he held the film close to his heart, in the hopes of doing justice to his son's illness.
When Couric asked De Niro if he felt "responsible" for a role the director was "personally invested in," De Niro stammered, fighting back tears.
"I don't like to get emotional," De Niro told Couric, reaching for a tissue. "But I know what [Russell] goes through."
Russell added that De Niro had also choked up during a meeting to discuss the film.
"I thought he was having hay fever, and I realized he was having an emotional reaction," said Russell.
"I sat there and watched Robert De Niro cry for 10 minutes and I said, 'Wow, he's really connecting with this material and this would be a beautiful thing if it could work out, because I think his heart would be there,' and it is there."
The exchange begins at around minute 5:00 in the video below:
Jennifer Lawrence, who stars alongside De Niro and Cooper in the film, mentioned Russell's son in her Screen Actors Guild Award acceptance speech.
"You made a movie for your son," she said. "So that he wouldn't feel alone and so that he could feel understood."