Billups also said his concern over the amount of assets in the franchise if LeBron James leaves in free agency next summer contributed to his decision to pass on the job.
"[Irving's trade request] didn't surprise me. I knew as [the Cavs] were doing their due diligence on me I was doing the same thing on them," Billups said in an interview on the Vic Lombardi Show on Altitude Sports 950 AM in Denver on Tuesday.
"I knew so much about the situation that the rest of the world doesn't know. But it's unfortunate, man, because [Irving is] a special talent. So much of what he's been able to accomplish on and off the floor has been as the beneficiary of having LeBron James."
Billups, an NBA analyst for ESPN, said he's surprised Irving would want a trade away from James.
"You're getting everything you want," Billups said. "You get all the shots you want, you're playing for a great coach who's letting you go to work, when the game is on the line they're coming to you, you're playing on TV every week. To me, I don't get it. Everybody has their own desires.
"Maybe he wants to be Russell Westbrook and go try to win the MVP and get all the shots. That's the only sense I can make of it, and to me that doesn't make sense."
Later on Tuesday, Billups said on SportsCenter that he was not aware that Irving was planning to request a trade.
"It wasn't that I knew Kyrie wanted to be traded or not, or him and LeBron had a beef," Billups said. "I don't know any of those things."
Billups didn't speak with James until after he decided to turn down the job, and said James' plans for 2018 when he can become a free agent didn't directly factor into the decision.
"The whole LeBron leaving the next year, to be honest that didn't bother me that much," Billups said during the earlier radio interview. "Here's why: When you have an opportunity to really put something together and put your imprint on it -- rebuilding is a beautiful thing if [they're] going to have the patience with you. What bothered me more than if LeBron left or not was that I didn't think they had great assets if you have to do a rebuild. It was more that than Bron. Bron and I have always had an amazing relationship."
Billups said the two-week process interviewing with Cavs owner Dan Gilbert was positive. He withdrew from consideration for the job on July 3. Reportedly, Gilbert offered him an initial contract of less than $2 million per season for the role of president of basketball operations.
"It was a great process. That team is a great team, a lot of expectations, lot of pressure involved with that team. But I didn't mind that. That's kind of been my entire career, the stress and pressure," Billups said.
"At the end, in my spirit, it just didn't quite feel like it was the right time at the end. It was one of those gut things. So I took myself out of consideration. I felt good about it. Dan and I talked afterwards and I thanked him for giving me such a good opportunity. ... We will remain friends.
"I don't need the job. I love my life. I love this ESPN thing; it's been fun. And I'll just wait until it feels right."