The highly-rated forward has been linked with Scottish Premiership duo Celtic and Rangers but could now be handed a Premier League move by the Blades.
Campbell, who is 20 years old, is out of contract at the Bet365 Stadium at the end of the season and has been offered a new deal by the Potters. However, he is believed to be weighing up his options.
He scored the only goal of the game last night as Stoke stunned top of the table West Brom at the Hawthorns to further enhance his reputation.
The youngster has enjoyed life under new Potters boss Michael O’Neill as the ex-Northern Ireland man has given him more opportunities. He has made 22 appearances so far this term in all competitions, chipping in with five goals.
Campbell joined Stoke in 2016 from Manchester City for a fee of around £1.75 million and made his debut in a Premier League fixture against Leicester City in February 2018.
He was loaned out to Shrewsbury Town last January for the remainder of the 2018/19 season and scored five goals in 15 games for the League One side.
Sheffield United are hoping for a couple of acquisitions before the end of the transfer window and looking to beat Celtic and Rangers to his signature. Stoke are facing a major battle to keep hold of him and could be resigned to losing with his uncertain contract situation.