Twiddling a pen, Harry tells Obama: "You're excited about this, I'm nervous about this, that's what's quite amusing".
"No no, let's keep it this way", Harry says. "I'd much prefer that", Prince Harry said before chuckling.
Obama also reminisced about his memories of leaving office and discussed his post-presidential work with the Obama Foundation, according to the palace.
Harry was interviewing Barry as part of his guest editorship of the programme, recording the interview during the Invictus Games in Toronto in September. Prince Harry interviewed President Obama at the Invictus Games. The duo also exchanges a couple of jokes.
Today tweeted a video clip showing Harry and Obama warming up ahead of the chat, with the former president joking: "Do I need a British accent?"
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The airport terminal was also momentarily evacuated, while trains to and from Schiphol were cancelled, local media reported. A spokesman for the Dutch police said: "the circumstances are telling us that it has nothing to do with terrorism".
"No, but if you start using long pauses between answers you're going to get 'the face, '" Prince Harry replies, demonstrating the stern scowl he's referring to.
When Harry showed him what he meant, Obama said: "I don't want to see that face".
The Prince and the ex-President appear to have developed a friendship beginning when Mr Obama was in office.
Michelle even co-hosted the opening ceremony of the Invictus Games with Prince Harry when it came to Florida previous year.
Michelle and I are delighted to congratulate Prince Harry and Meghan Markle on their engagement.