Rodriguez is thought to be open to a move this summer following West Brom's relegation from the top flight, with former club Burnley known to hold a serious interest in him and 'Baggies' team-mate Craig Dawson.
According to the aforementioned source, the 'Clarets' have made their interest in the England international clear, lodging an official £12m bid which West Brom have rejected out of hand.
The 28-year-old started his career at Turf Moor, coming through the academy and scoring 41 goals before being snapped up by Southampton for £7m in 2012.
Such was his form on the South coast that Rodriguez earned himself and England call-up, however, after missing almost a year of action due to a knee injury, he failed to regain his place in the team at St Mary's, before joining West Brom for £12m last summer.