Medvedenko, 43, was a power forward on the Lakers' championship teams in 2001 and '02, playing alongside Kobe Bryant and Shaquille O'Neal.
SCP Auctions is donating the entire final sale price of both rings to Medvedenko's Fly High Foundation. Its goal is to support Ukrainian children by restoring the sports infrastructure of the war-torn country's schools and launching a network of social sports clubs.
"We want to restore gyms because the Russian army bombed more than a hundred schools," he told The Associated Press by phone on Sunday. "Our country, they need a lot of money to fix the schools. Sports gyms are going to be last in the line to fix it. In Ukraine, we have winter and kids need to play inside."
The auction runs from Wednesday through Aug. 5. The Laguna Niguel, California-based company estimates both rings will raise at least $100,000.
Medvedenko said he decided to sell the rings after going to the roof of one of the tallest buildings in his Kyiv neighborhood and watching rockets launched by Russian forces streak through the night sky.
"In this moment I just decided, 'Why do I need these rings if they're just sitting in my safe?'" Medvedenko said. "I just recognize I can die. After that, I just say I have to sell them to show people leadership, to help my Ukrainian people to live better, to help kids."
Medvedenko spoke from Warsaw, Poland, where he staged a sold-out charity basketball game to raise money for Ukrainian refugees who crossed the border to escape the war.
"In Ukraine, you're just feeling it's war, rockets, air alerts. You're so used to that kind of pressure," he said. "As soon as you cross the border and see how people live normal life, it's a different world."
Medvedenko is married with two daughters, ages 16 and 11, and a 10-year-old son. After Russia invaded Ukraine in February, Medvedenko sent his children to live with their grandmother in another part of the country.
"After they stay for 1½ months, they were all the time calling me and asking, 'Papa, can we come home? We want to be with you and Mama,'" he recalled.
Five months into the war, Medvedenko has reunited his family in Kyiv.
"We have air alerts almost every day. Sometimes it's three or four times a day," he said. "The kids are so used to it. They play in our backyard. They not even stop playing they are used to it."
Medvedenko has served in Ukraine's territorial defense forces during the war.
"We were defending our neighborhood, doing checkpoints and duty patrol. I'm not the best solider, I'm not the best shooter, but I can give them support," he said, adding that he carried an AK-47. "I shoot it a couple times, not at people. I'm happy I don't have a chance to shoot somebody. Our army did a great job to defend Kyiv. I want to thank them."
Medvedenko was a candidate for Kyiv City Council in the 2020 election. He was 11th on the election list and his party only managed to win nine seats.
Beyond his humanitarian efforts during the war, Medvedenko has long-term goals to help his country.
"After the victory, we will definitely return to that question of quality changes in sport," he said. "Ten years in the United States, I saw how it works. I hope I have ideal model in my mind to change Ukrainian sport."
Medvedenko joined the Lakers in the 2000-01 season. He had his best season in 2003-04 when he started 38 games in place of injured Hall of Famer Karl Malone, and averaged 8.3 points and 5.0 rebounds. Injuries later slowed him and Medvedenko was traded to the Atlanta Hawks in 2006-07, his final season in the league.
Medvedenko said he texts with former Lakers Mark Madsen and Luke Walton. The team has sent sports equipment for use in Ukraine.
"The Lakers family always help me," he said. "The Lakers are always in my heart."