The left-footed player, who can play in either midfield or as a wingback, has attracted interest after a series of eye-catching performances for the Robins last season, according to The Sun.
Rafael Benitez is looking to add competition for Paul Dummett in defence, and the 24-year old's relatively low price tag of £6m is understood to have caught his attention.
Bryan has also attracted interest from Germany, with clubs such as Augsburg and Mainz 05 keeping tabs on the player.
Premier League duo Bournemouth and Wolverhampton Wanderers and fellow Championship sides Leeds United and Middlesbrough are also thought to be interested, although Bristol are unlikely to sell to their rivals.
Bryan made 48 appearances last season, scoring seven goals.