Rodriguez suffered the injury in Southampton's 4-1 loss against Manchester City on Saturday and his club announced on Twitter that he would now be sidelined "for around six months".
It brings to an end any hopes the 24-year-old had of being selected in Roy Hodgson's England squad for this year's tournament in Brazil, which he had put himself in contention for after scoring 15 goals in 33 Premier League games this season.
A club statement revealed that Rodriguez had already met with a knee specialist and that he would undergo a "routine operation shortly".
Les Reed, head of football development at St Mary's Stadium, insists the most important thing is for the former Burnley man to focus on making a full recovery.
"Jay is naturally disappointed, but is determined to get back playing for Southampton as soon as possible," Reed said in a quote tweeted by the club on Monday.
"Jay has asked us to pass on his gratitude to everybody who has shown him their concern and sympathy.
"Everyone at the club now wishes Jay well and we will support him, along with his family."
Rodriguez earned his first England cap in November last year, starting in a 2-0 defeat against Chile at Wembley.
The same injury has already ruled Arsenal winger Theo Walcott out of the World Cup.
