The forward will be out of action for six months with a knee injury.
The 24-year-old ruptured his anterior cruciate ligament after landing awkwardly during the Saints' 4-1 defeat to Manchester City last weekend.
Despite Rodriguez being ruled out of this summer's World Cup in Brazil, Pochettino has insisted that he is positive mentally.
"We're all very sad for Jay, but we know he's a very strong character," Pochettino said in Friday's press conference. "He's an optimist and he's doing well mentally.
"I am certain that Jay will become stronger from this situation. I'm positive that he'll return reinvigorated. Jay has been coming to the training ground every day, and he's started his recovery already. He's in high spirits.
"At the moment, the medical team is working to bring Jay's swelling down and then will decide when is best for his surgery."
Southampton will take on Cardiff City at St Mary's on Saturday.