The 18-year-old, who has made three League Two appearances for the Shrews last season, is out of contract and, according to his agent, will not be returning to the club meaning he is free to sign for other clubs.
A compensation package would be required, as was the case for us with Chris Long, one that was agreed with Everton, and, in the other direction, Danny Ings and Liverpool, which the two clubs failed to agree on.
Shrewsbury chief exec Matt Williams said: "Burnley made us an offer but we turned it down and told them what we felt we'd get at a tribunal. They have left their offer on the table and gone away to look at their next step.
"His agent told us he definitely wouldn't be signing for us. His agent also reminded us Josh could go abroad or play in the Conference North for a year, but that's a risk we're willing to take."
Burnley could now up their offer and agree a deal, sign him and accept the tribunal figure, or, alternatively, look elsewhere.
Update:
Further comment from Matt Williams: "I'm not disclosing how much the offer was. Burnley are doing what they think is right for their development group, but we've done some due diligance and research into similar cases that have gone to tribunal and we know where we should be and what we should be getting and we've told that to the agent and Burnley.
"We have no issue with Burnley, they're a great club and I know their chief executive (David Baldwin) from his time at Bradford. We've had a couple of open discussions with them and we'll remain in dialogue with them."