The 15 year-old is joining Newcastle ahead of rivals Manchester United and Tottenham.
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Sky Sports says a £350,000 offer from Newcastle has been accepted by the Tykes.
Kenchington has been likened to Manchester City defender and former Barnsley prospect John Stones.
The teen has confirmed his exit on Instagram, writing: "Where do I start. five years at this great academy, filled with both ups and downs, a rollercoaster of emotions.
"I would like to take this opportunity to thank everyone at the club, including academy staff, players, and every coach along the way, thank you all.
"I will never forget the memories made at my boyhood club and never forget how much I progressed. But it's time now for the next chapter of my life. We move.."