The 24-year-old is now discussing personal terms over a permanent move to the Championship side.
Barrow made a positive start to the season under Francesco Guidolin and then Bob Bradley, but fell out of favour when current boss Paul Clement joined the club in January.
The winger went on to spend the second half of the season on loan at Leeds, where he played under former Swansea manager Garry Monk.
Swansea are now ready to let the Barrow leave and kick-start his career at Reading.