Nigel Adkins' Barclays Premier League newcomers had an offer rejected for the 22-year-old 12 months ago but are understood to have come closest to matching the Clarets' £8million valuation.
Rodriguez enjoyed a blistering campaign last term, netting 21 goals despite having his contribution limited from early March by hip and groin problems.
The former England Under-21 international is currently out of the country and is yet to meet with Saints officials to discuss personal terms.
Should the deal go through, it is set to surpass the £6.5million recouped by Burnley for the sale of Scotland striker Steve Fletcher to Wolves two years ago.
Burnley declined to comment on the story when contacted by Press Association Sport.