The forward's hopes of representing England at his first World Cup are over after he was stretchered off following an awkward landing during Saturday's game
Rodriguez was stretched off after just 26 minutes at the Etihad Stadium after landing awkwardly and scans have now confirmed the injury with the 24-year-old set to undergo surgery on the problem shortly.
The injury means the 24-year-old will not feature again this season for the Saints and will also miss the tournament in Brazil after recently breaking into Roy Hodgson's England squad.
"Jay is naturally disappointed, but is determined to get back playing for Saints as soon as possible," said club director Les Reed.
"Everything will be done to return him to full fitness and deliver a speedy recovery.
"Jay has asked us to pass on his gratitude to everybody who has shown him their concern and sympathy during the 48-or-so hours since his injury.
"Everyone at the Club now wishes Jay well and we will support him, along with his family, to ensure that an international career plays a big part in his future."
Rodriguez has scored 15 goals this season for Southampton and made his England debut in November, in a friendly defeat against Chile.