According to Football Insider, the Clarets, as well as Fulham and Swansea City, are interested in signing the defender, who is out of contract with the Blades at the end of the season.
As recently as February, The Star reported that talks over an extension to his deal at Bramall Lane were yet to get underway, and with no indication that a new deal is any closer, it would appear that there is now a race to acquire his services.
Bryan came through the ranks at Manchester City, but failed to make a senior appearance for Pep Guardiola’s side before leaving to join Sheffield United on a free transfer in 2018.
A defensive injury crisis initially thrust the 24-year-old into the first team Bramall Lane in November, and he has gone on to feature 10 times in the Premier League since then, scoring once in the shock 2-1 victory over Manchester United in January.
The biggest hurdle that Burnley may have to overcome, however, is that Bryan himself has previously admitted that he would ideally like to stay with the Blades this summer.
Elsewhere in the Premier League, Brighton and West Brom are both said to be keen on Standard Liege teenager Michel-Ange Balikwisha, according to Le Soir.
The 19-year-old has scored seven goals and assisted four more this term, and is described as an “elegant” talent who is in no rush to pen a new deal with his current employers.
We’ve gathered the rest of today's Premier League speculation below...