The 37-year-old former NBA player admitted to being unfaithful to Kardashian while discussing his drug use and hospitalization in the April 10 issue of Us Weekly.
"If there is one thing I regret when I was married, it was having multiple affairs with different women. That wasn't the stand-up thing to do," he told the magazine.
Kardashian initially filed for divorce from Odom in 2013 after four years of marriage. The former Los Angeles Lakers star said in the interview that Kardashian had discovered his cocaine use about two years prior.
"I was hiding it for a while ... [but] she caught me. She was disappointed. So was I. The sad thing is, I don't know if I was disappointed because I was actually doing the drug or because she caught me," he confessed.
"She knew I was doing cocaine the whole time after that," the star claimed. "It was my drug of choice. I'm not going to say she accepted it because that would be the wrong word. Tolerated would be a better word."
Kardashian put the divorce on hold following Odom's highly-publicized hospitalization in 2015. The former NBA player acknowledged he was discovered unconscious in a Las Vegas brothel, but denied the incident was drug-related.
"Looking back, I might have had a drink to get the mood started, but was I drunk or on drugs? Not at all," he asserted. "I remember lying in bed. Two women were in bed and then I fell asleep. That's all. When I woke up four days later, I was trying to pull the tubes out of my mouth."
Kardashian supported Odom throughout the ordeal, but ultimately re-filed for divorce and finalized the process in December. The reality star previously said in an essay for Lenny Letter that she was "at peace" with the decision.
"[I] do feel like I honored my vows to the very end," she said. "I feel that I kept the serious vows I made in front of God with every inch of my heart, which is why I am still honoring them today even though my relationship with my ex-husband is in a different place."