The Clarets are keen to take the winger to Turf Moor to link up with their development squad and tabled a second offer.
But the Sky Bet League 1 Shrews have turned it down and will not settle for less than their asking price.
Ginnelly, 18, has been training with Burnley and will not sign a new deal with Shrewsbury but chief executive Matt Williams confirmed they have dismissed another bid.
He said: “We had two offers towards the end of last week. The first offer we rejected, as you would if someone was trying to buy your house. The second offer was a very slight increase on the first which we also rejected. They have left that on the table. We have explained to them our position.
“It’s very amicable, we have said to them we think we’d get more at a tribunal and told them what we’d be looking for.
“I think they are reluctant to get involved in a tribunal, they have one going on with Liverpool for Danny Ings – it’s a shame we’re not talking about the same sort of figure.
“I will probably have another conversation with Burnley in the next 24 or 48 hours.”
Ginnelly has made three substitute appearances for Shrewsbury, all last season, as they won promotion from Sky Bet League 2.