The League One champions last week had an opening offer rejected and their improved return, reported to be in the region of £1.5million with add-ons, has also been rebuffed by the Blades.
"We've had one or two bids that have been rejected so far for Harry. Both bids have come from the same club. He has the chance to play Championship football next season," United managing director Mal Brannigan told the Express & Star.
Maguire has a year remaining on his contract at Bramall Lane and Blades boss Nigel Clough has made no attempt to disguise the fact that he believes the 21-year-old would be best served by staying for at least another season.
Clough has made the player and improved contract offer but, as yet, that remains unsigned.
Brannigan admits United could be tempted to cash in, adding: "If you look at the amount of money that could come in for Harry, we could utilise it on the squad, but we'd much prefer him to stay.
"It's very much in his interests to extend his contract which we'd like him to do and there's an offer on the table for him.
"If he doesn't, we'd like him to stay for the remainder of his contract. I think for his career, there's value in seeing this one out so he can hopefully see us through to the Championship.
"But Harry wouldn't be the first player to come through a club's academy and then want to play elsewhere."