The 22-year-old was previously linked with a move to the Ibrox club last summer but no deal transpired.
Rangers added a number of options in the wide areas this summer, with new boys Sheyi Ojo, Greg Stewart, Jake Hastie and Jordan Jones all capable of playing on the flank.
But with Daniel Candeias having left and Glenn Middleton expected to depart on loan - possibly to Hibs - Steven Gerrard is still keen on bolstering the area.
Barker impressed during a loan spell at Easter Road in the 2017/18 season, playing 27 times for Neil Lennon's side.