The 24-year-old Dutch midfielder, who joined from Feyenoord earlier this summer, had to be substituted due to a hamstring injury in Thursday's 3-0 Europa League qualifying win over Vitesse at St Mary's.
"Clasie had some hamstring problems - we will find out more tomorrow," Koeman said.
"It is possible he will miss the start of the season because hamstring injuries are always dangerous." The Saints will kick-off their Premier League campaign against Newcastle United on August 9.
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