It has been reported that the Gers want to secure the services of the former England Under-20 international on a season-long loan deal.
The 21-year-old winger spent the 2017-18 campaign on loan at Hibernian from Chelsea last season, and scored two goals and provided five assists in 27 Scottish Premiership appearances, according to Transfermarkt.com.
Former Rangers striker Kenny Miller, who left the Gers this summer to take charge of Scottish Premiership side Livingston, has spoken highly of Barker and was impressed with his performance at his own former club Hibs.
Miller told The Scotsman: “I thought Brandon Barker was excellent at Hibs last season as well. By the looks of things, he might move on to other things this summer on the back of a successful loan spell.”
Rangers have been quite busy in the summer transfer window and have already made a number of signings.
The Gers will be aiming to challenge Old Firm rivals Celtic for the Scottish Premiership title next season.