The club are continuing their rebuild under manager Steven Gerrard and have pinpointed the 21-year-old, who would add pace and excitement to the team.
Gerrard and director of football Mark Allen have put together a long list of players who can help Rangers progress next season and close the gap on Celtic.
Barker is one of those names, but Rangers will likely face competition from a host of Championship clubs who are interested in the speedster, as well as Hibs who are desperate to bring the player back to Easter Road.
The Englishman proved a key asset to Hibs with his ability to play on both wings, or more centrally. He got fans out their seats with his pace and dribbling ability.
He made 30 appearances scoring twice last season having previously spent time on loan at Rotherham and Dutch side NAC Breda.