According to Football Insider, Burnley, Fulham, and Swansea City are all interested in signing the defender, who is out of contract at the end of the season.
As of February, talks over an extension to his deal at Bramall Lane were yet to get underway, and the uncertainty over his future has seemingly sparked a scramble to acquire his services.
Bryan joined to the Blades in 2018 after leaving Manchester City's youth team setup, and has featured 10 times in the Premier League so far this season.
The player himself has stated previously that he wishes to stay at Bramall Lane, although it remains to be seen whether he would resist the lure of a chance to stay in the top flight should United succumb to relegation.
Bryan scored his first ever Premier League goal in the shock 2-1 win over Manchester United towards the end of January, and has made a big impression since coming into the side back in November.
Elsewhere, Sheffield United will not be deterred by Slavisa Jokanovic’s high wage demands should they decide to make a move for him.
The Serbian manager is said to be one of the leading figures in the race to replace Chris Wilder at Bramall Lane, but securing his services won’t be straightforward.
The Sun report that Jokanovic is currently on a salary of around £2.5 million per year in Qatar, and that the Blades would need to match that figure if they were to lure him back to the English game.
But according to Yorkshire Live, that sum will not be enough to put off the Blades, who are more than willing to table a competitive offer.
The 52-year-old has good experience of life in England, having managed 180 games over spells at both Fulham and Watford.
Since July 2019 he has been in charge of Al-Gharafa SC, where he has averaged 1.58 points per game.
In nearly three years at Fulham, he averaged a PPG of 1.56.
We’ve gathered the rest of today's Premier League speculation below...