The bid is understood to have been £100,000, an amount well below the Paisley club's valuation of the 19-year-old.
McGinn made 38 appearances for St Mirren last season - scoring three times - having helped the club to win the League Cup in his breakthrough campaign a year earlier. He also earned his first cap with Scotland Under-21s in March.
St Mirren rate the teenager highly and are thought to be unimpressed by the offer made from Bramall Lane as a result. "The board will not stand in any player's way if the price is right but in this case, it was nowhere near right. The word they are using is derisory," said a club source.
"The club wants to keep all it's good players but knows there comes a time when they have to move on. But that will only happen if they receive fair value."
United would be able to offer more than money to McGinn if a fee is agreed, since his older brother, former St Mirren midfielder Stephen, is already a player at Bramall Lane. The Sky Bet League 1 club also signed Scottish striker Marc McNulty from Livingston this month, while compatriots Ryan Flynn, Jamie Murphy, Stefan Scougall, Bob Harris and Neill Collins are all United players.
Nigel Clough, the United manager, has proven to have a keen knowledge of the Scottish scene having taken such as erstwhile Dundee United striker Johnny Russell to Derby County before his appointment in Sheffield. It is a market which the coach is determined to pilfer further.
"We've always maintained that Scotland is a good market for us because, given the money situation, young players are given an opportunity quickly," added Clough.