Psy's song "Gangnam Style," which mocks the consumerism of a rich Seoul suburb, and hit horse-riding-style dance went viral on video-sharing website YouTube. It has been viewed more than 530 million times on YouTube since it was released in mid-July.
"I'm a bit jealous," Ban told reporters. "Until two days ago someone told me I am the most famous Korean in the world. Now I have to relinquish."
"We have a tough negotiations in the United Nations. In such a case I was also thinking of playing 'Gangnam Style' dance so that everybody would stop and dance, maybe you can bring U.N. style," he said.
But while he briefly posed like he was doing Psy's trademark dance, Ban wasn't ready to attempt it.
"I know that you are here to see me dance but don't worry - I cannot even imitate your movement," Ban said.
Ban earlier on Tuesday met Baumgartner, who last week leapt into the stratosphere from a balloon near the edge of space 24 miles above Earth, setting a record for the highest skydive and breaking the sound barrier in the process.
Ban asked Baumgartner if he would be able to jump from the top floor of the United Nations headquarters, to which the skydiver replied: "With your permission, sure." But he later added: "I'm officially retired from the daredevil business now."
Ban hailed Baumgartner - who has made a career of risky jumps including skydiving across the English Channel and parachuting off the Petronas Towers in Malaysia - as "the most courageous person in the world."
While the U.N. secretary-general deals with weighty issues such as the conflict in Syria and crisis in Mali, he also said he believes it is important to engage other parts of society.
"I think the music can play a very important role. I hope that we can work together using your global reach. ... You have, I think, unlimited global reach," he told Psy.
"You are so cool, I hope that you can end the global warming," Ban added.
(Reporting by Michelle Nichols; Editing by Will Dunham)