The 24-year-old has scored six goals and assisted three others in 47 games for the Championship club this term, while he also played a huge role as they reached the EFL Cup semi final.
Bryan can also play further up the pitch in midfield, and such versatility has attracted the Clarets.
Premier League football will no doubt tempt the fullback, and Bristol City won't likely hesitate to let him go if they deem the price right.
However, their fans want Burnley to increase their reported initial offer given the quality Bryan has displayed this term.
They can be ready to have that rejected as well. Not even close to enough.— JC (@JC_GreenArrow) May 2, 2018
Lol. Only 6m. Double it and not even close.— Sid J (@sidlondon) May 2, 2018
A simple 'no' will do, double figures please Sean— Chris Lear (@chrisjlear) May 3, 2018
15mill and they might get a deal— Steve Moore (@SteveMo36) May 2, 2018
He's worth a lot more than £6m if City let him go for that small fee somebody needs sacking!!!!— Mark carroll (@Markcar24821976) May 2, 2018
The Turf Moor outfit are looking to bolster their squad in preparation for additional commitment next season, with Europa League football all but certain.
With Charlie Taylor failing to provide 32-year-old Stephen Ward strong competition for playing minutes, Bryan has been recognised as the man capable of doing exactly just that.
Burnley paid Leeds United around £6million as compensation fees for Taylor's signature last summer, and Bristol City will surely demand more from them for Bryan's services.
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