The 18-year-old has agreed a five-year contract after turning down an offer from Hibs.
It is believed Celtic paid £125,000 and that the deal includes add-ons and a sell-on clause.
Kenny is their fourth signing of the January window following the arrivals of J-League trio Daizen Maeda, Reo Hatate and Yosuke Ideguchi.
Manager Ange Postecoglou has not ruled out making more.
"I don't expect it to be too hectic but we'll stay alert and make sure that if the right thing comes along both in terms of incomings and outgoings we'll be able to move," he said.
It is likely Kenny will join the Celtic B team in the first instance and ply his trade in the Lowland League while working with the Celtic coaching staff to develop his game.