The Saints snapped up the one-cap England international from Burnley upon their Premier League ascension back in summer 2012.
He enjoyed an emphatic 2013/14 campaign, netting 15 times in 33 appearances and pushing for a slot in Roy Hodgson’s World Cup squad, until suffering a serious ACL injury in April.
That has seen the 27-year-old miss the vast majority of the last two seasons and although he was on the bench for Southampton’s first match of the current campaign, it’s unknown whether he features prominently in Claude Puel’s plans.
Hull City, meanwhile, are in the market for loan acquisitions because of their impending takeover, which makes permanent transfer deals more risky.
The Tigers have reportedly made an approach to sign Rodriguez for the season, but a Blackburn switch for fellow forward Sam Gallagher could complicate the potential deal.