The 27-year-old has suffered a succession of knee injuries in the past few seasons and the Saints are hoping to loan the forward out so he can regain form and fitness.
West Brom are currently in pole position to sign Rodriguez, although Hull has also been touted as a possible destination for the Southampton man.
Rodriguez remains a popular figure at St. Mary's and it is understood that Southampton do not want the player to leave on a permanent basis.
The former Burnley forward has made 76 league appearances for Southampton, scoring 21 goals and also has one England cap.