He was midway through answering the presenter's questions about Park Guen-hye when a little girl, dressed in a yellow jumper, is seen opening the door behind him and wandering into the room.
She swings her arms happily as she walks over to her dad, who has clearly spotted her on his screen and attempts to push her out of the way.
The broadcaster continues to ask his next question while pointing out: "... I think one of your children has just walked in."
Meanwhile a younger child in a baby walker comes waddling into the now open doorway and follows after her sister towards Mr Kelly, a professor of political science and international relations based in Busan, South Korea.
Seconds later, a woman - thought to be their mother - comes bounding into the room and grabs both children by their arms, dragging them backwards.
In her hurry she knocks books off the nearby table before banging into the door which had slammed behind her.
The older child can be heard shouting: "Oww, mummy!"
Their dad tries to remain professional but closes his eyes for a few seconds and apologises as they make their dramatic exit.
He continues to answer the question while in the rear the mum can be seen on her hands and knees quickly trying to shut the door.
A clip of the interview was posted on Twitter and has quickly gone viral.
Lee Sherry wrote on Twitter: "Did anybody just watch BBC world news interview regarding Korean president... children walked in. Maid dragged children out? Hilarious".
Tony Brown said: "I just can't stop watching it on repeat."
Afterwards Professor Kelly was asked if he minded the video being shared by a BBC team member on Twitter.
He responded: "What would that mean, please? Re-broadcasting it on BBC TV, or just here on Twitter? Is this kinda thing that goes 'viral' and gets weird?"
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