Speaking in an interview with ESPN's Colleen Dominguez, White said he was going to meet with general manager Daryl Morey on Monday to discuss his situation with the team.
"I'd rather tell them on the front end and be honest and transparent and never play again for that than allow me to become one of the stories because I wasn't able to communicate," White said.
White has remained absent from the team in a dispute over how the Rockets are helping him confront his anxiety disorder.
"We all look at the stories that happen later and go, 'Man, you just wish this guy was able to communicate his problems,' " White said. "Or you wish somebody would just've talked to him and wished that the communication would've been there.
"Well now the communication's there and there's still a problem, right? That means the problem isn't us. The problem is the art of the business, right? At no point will I compromise my health in the interest of business."
Asked if he was sure he would give up his NBA career in the interest of openness and honesty, White replied, "If that's what it means."
The Rockets have said they intend to fine White for every day he remains away from the team or does not attend sessions with a therapist arranged by the team. The Houston Chronicle has reported the Rockets have no plans to trade or release White.
The Rockets said Friday night the team stands by its previous statement regarding White.
"We are committed to his long-term success and we will continue to support him now and going forward," the team said in a news release.
Meanwhile, the Harlem Globetrotters have reached out to the rookie, who publicly has stated his fear of flying.
"I have watched from afar an incredibly gifted basketball player who has struggled off the court with a fear of flying, and I truly feel for this young man," Globetrotters coach and director of player personnel "Sweet" Lou Dunbar said in a news release. "While we truly hope Royce can overcome his fears, and have a highly successful NBA career, I just want him to know the Globetrotters offer their support and another option to play professional basketball without a rigorous flying schedule.
"He could have an opportunity to play for the Globetrotters, as many of our 270 games in North America are traveled by luxury bus. As for the 150 others overseas, well, we would have to excuse him from those games."
White has missed practice and did not attend the Rockets' three games this week, including a 119-117 overtime loss at Portland on Friday night, which dropped them to 4-5. He earlier wrote on his Twitter account that the Rockets have been "inconsistent" in helping him.
Information from The Associated Press was used in this report.