The Tigers are ready to make an approach for the 27-year-old forward, who featured in just 12 top-flight games for the Saints last term with the ex-Burnley player on the sidelines for the 2014-15 campaign with injury.
The Tigers want to secure the services of Rodriguez on a season-long loan deal.
Hull are in the market for a number of players and they have a bid lined up for Hungarian Nemanja Nikolic, but Polish side Legia Warsaw are stalling on the sale.
Sheffield United are aiming to complete the signing of Cardiff City goalkeeper Simon Moore before tomorrow’s game at Millwall.
The 26-year-old is out of favour in South Wales and a deal of around £500,000 has been agreed with the goalkeeper undergoing a medical last night.
Moore was signed from Brentford in 2013, but has found his first-team opportunities limited by the form of Scottish international David Marshall.
Moore made six league starts for the Welsh outfit last season.
Huddersfield Town full-back Jason Davidson has joined Dutch side FC Groningen on a season-long deal.
The 25-year-old has been on the periphery with Terriers head coach David Wagner since last season and the Eredivisie outfit have an option to make the deal permanent.
On Davidson, contracted at Town until 2018, said: “This is a good move for everyone.
“Jason is a senior player and is obviously keen to play regular football and that is not something we can offer here at the moment.
“He has a profile in the Netherlands from his time playing in the country and we all wish him the best of luck.”