The 23-year-old was forced off the field in the Saints' 2-0 win over Sheffield Wednesday in the League Cup earlier this week after a lengthy spell on the sidelines.
Cork tweeted: "Another 4-6 weeks with my left knee. Lucks just not going my way this season. Could be worse, but was so close to getting back from the ankle."
Cork was competing in his first game for the former Championship club since picking up an ankle problem before the start of the Premier League campaign.