Rodriguez joined the Saints from Burnley for £6million in June 2012 and has scored 26 goals in 76 appearances.
That form earned him a place in the England squad for the friendlies against Chile and Germany last year and he made his international debut in the 2-0 defeat to the South Americans in November 2013.
Southampton have already lost midfielder Adam Lallana and Rodriguez’s fellow striker Rickie Lambert to Liverpool during the close season.
England left-back Luke Shaw has also left St Mary’s to join Manchester United in a £27m deal but Southampton appear keen to retain one of their other emerging talents.
Watch Southampton play Liverpool at Anfield live on Sky Sports on August 17 as part of the coverage of the opening weekend of the Premier League season.