Twiddling a pen, Harry tells Mr Obama: "You're excited about this, I'm nervous about this, that's what's quite amusing".
The interview with Mr Obama was part of Prince Harry's guest editorship of the BBC's Today program.
Prince Harry is making dreams come true once again. Mr Obama asked during the clip, adding, "Do I need a British accent?"
"If you start using long pauses between your answers, you're going to get 'the face, '" Harry jokingly warned in response.
Obama, meanwhile, offered to interview Harry, who declined.
Mr Obama laughed: "I don't want to see that face".
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"No, no. Let's keep it this way".
The interview was recorded during the Invictus Games in Toronto in September, according to a tweet from BBC Radio 4 Today and will be broadcast two days after Christmas.
The palace said the conversation features Obama discussing his plans to cultivate the next generation of leadership through the Obama Foundation.
The prince and the president have enjoyed a friendship since Obama was in office.
At this year's Invictus Games, Obama reportedly quizzed Harry on his relationship with Meghan Markle as they watched wheelchair basketball.
Kensington Palace released a preview on Sunday of what appeared to be a sound check.