The Buddies announced on their website that they had knocked back an approach from an unnamed club for the Scotland Under-21 international midfielder.
But reports suggest the offer came from Bramall Lane, where older brother Stephen is currently on the books.
However, the bid - reported to be £100,000 - has angered the Paisley club.
"The board will not stand in any player's way if the price is right but in this case, it was nowhere near right," a source close to the club said. "The word they are using is derisory.
"The club wants to keep all its good players but knows there comes a time when they have to move on. But that will only happen if the receive fair value."
The website statement confirmed they had rejected the offer for the 2013 League Cup winner.
It read: "Following media speculation regarding player John McGinn, we can confirm that we have received an offer for John which was well below our valuation of the player and was rejected immediately."