According to the Express, the former Liverpool starlet has been told he can leave Ibrox, and Terrors boss Robbie Neilson is understood to be interested in taking the 21-year-old on loan as he chases promotion from the Ladbrokes Championship.
However, Barton - now manager at Fleetwood - is also keen on Rossiter. The Trawlermen currently lie tenth in League One, but are just eight points off the relegation spots.
They have already taken Jason Holt on loan from the Gers, and Barton is now thought to be keen on adding Rossiter to his squad.
However, Rangers would have to pay a sizeable chunk of his wages if a move is agreed.
The former England Under-21 cap has been plagued by injury since swapping Anfield for Ibrox, and has made just 16 appearances for the Light Blues since joining in July 2016.
With Rangers having added Finnish international midfielder Glen Kamara on a pre-contract deal from Dundee, opportunities for Rossiter would appear to be limited at Ibrox.
Meanwhile, Carlos Pena has interest from two Mexican clubs while Rangers are monitoring Larne midfielder Liam Hassin. The Northern Ireland Under-19 cap has caught the eye for the Harbour Rats and is on the radar of several clubs.