The Magpies are reportedly interested in signing the 23-year-old left-back.
Rafa Benitez has lost Paul Drummett to injury and had also made enquiries about Genoa full-back Diego Laxalt, but Bryan has emerged as a potential target for the Spaniard.
Dummett faces several weeks out of action after suffering a hamstring injury against Tottenham Hotspur during Newcastle’s opening Premier League match at St James’ Park.
Championship clubs Birmingham City and Leeds United have also been linked with a move for the 23-year-old, but The Robins will only sell him to a Premier League side.
Bristol have already rejected a bid from Birmingham City, while Leeds United are willing to pay up to £2 million to sign the defender.
City head coach Lee Johnson has insisted that the player will not be sold in the summer transfer window, but the Daily Mail reports that talks being held over a deal with Newcastle.
Bryan was ever present for Bristol City last season, and according to Who Scored, was statistically better than any other left-back in the Championship over the course of the 2016/17 campaign.
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