Acording to the 'Daily Mail', the 'Clarets' are keen on taking advantage of West Brom's perilous situation to bring the player they sold to Southampton in the summer of 2012 back to the club.
If, as expected, the 'Baggies' are relegated, they will be forced to cash in on several first-team stars, with Rodriguez among those who would be expected to depart.
The 28-year-old has been one of West Brom's better players this season, scoring 11 goals in all competitions to date.
Rodriguez came through the ranks at Burnley, scoring 41 goals in 128 games for the club before moving to the South Coast and his work ethic and affinity for the club, combined with Burnley's need to flesh out their squad ahead of a potential European campaign means a homecoming is possible.