Becker was released from a British prison in December 2022, having served eight months of a two-and-a-half-year sentence for hiding financial assets after he was declared bankrupt.
"I can confirm that I am Holger Rune's coach," Becker told Eurosport Germany's podcast.
"It makes me a little proud that he asked me. The contact has existed for a long time. Now it was a very good fit.
"My calendar allows it and I have always been interested in Holger because he is on the tennis court with so much commitment and temperament."
The German previously coached Novak Djokovic from 2013 to 2016, and will now train Rune.
The Dane reached the quarter-finals at Wimbledon this year but has won just one of his last nine matches.
"Holger is a rough diamond that needs polishing," Becker said.
"I like his emotional outbursts. I have coached a player before, Novak Djokovic, who was sometimes not quite himself on the court, but that is allowed."
Becker is currently barred from entering the UK before October 2024, and at the moment is unsure if his partnership with Rune will continue into next season.
"There's no point in signing a two-year contract now when you've only been cooperating for a week.
"If you are successful, you usually stay together, no matter what the contract says. If not, everyone goes their own way again."
