Ginnelly, 18, made three substitute appearances for Shrewsbury last term, following his debut against Luton Town.
After starting his career with Aston Villa's academy, Ginnelly joined Shrewsbury in 2013.
"I hope he goes on to be a huge success and shows the potential that he has," said Town boss Micky Mellon.
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Shrewsbury Town's teenage midfielder Josh Ginnelly has joined Championship side Burnley on a three-year deal for an undisclosed fee.