Piroe scored 19 times in the Championship last season and is one of the most wanted forwards in the second tier of English football.
Leeds have their eyes on the Dutchman but a few Premier League clubs have also been considering snapping him up.
Sheffield United are in the market for a striker and Piroe is a striker the club have been keeping tabs on him.
It has been claimed that they have probed the possibility of getting their hands on the Championship hitman.
But the Blades have baulked at the asking price Swansea have slapped on the forward this summer.
He is in the final year of his contract but Swansea are not interested in selling him at a cut-price fee.
It is unclear whether Sheffield United will decide to make a move to sign him towards the end of the window.
The Yorkshire club are also considering a move for AC Milan striker Divock Origi in the coming days.
