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Metcalfe, 20, has joined for an undisclosed fee and has signed a three-year contract.
"I am so happy to join the club and I am especially looking forward to spending at least three years here, hopefully," Metcalfe told Bradford's website.
Bantams boss Graham Alexander added: "Jenson is a player with big potential and will bring extra competition to our midfield.
"His attributes in and out of possession suit our game very well, so we look forward to seeing him integrate into life at Bradford City."