Kelly responded with a tirade of abuse on his Twitter account.
He claimed to have papers proving he had been given the band's permission to use the song.
"The band The Naked and Famous are the ones who cleared the song I sampled. I was paying hommage to them, not stealing, u f****n idiots," he said.
Soon afterward, The Naked and Famous deleted their original post.
The rapper then took to the band's Twitter account to continue challenging them over the allegation.
"Yall deleted your tweet? someone mustve showed u the paperwork of yall APPROVING that sample. Now apologize," he said.
Another post to the band said: "I was actually a huge fan of yours and was paying hommage to yall by making a play off the name to describe my situation."
He has since posted more tweets claiming to have found the paperwork proving the band had approved the song sample.
Management for The Naked and Famous did not respond to calls.