The midfielder, 31, suffered the problem during the warm-up on March 25 before the Saints' 1-0 win at St Mirren in the Scottish Premiership.
"Our season has already been marred by freakish, 'non-contact', serious injuries," Wright told the club's official website. "On a disappointing note, it looks like Gary may need a similar operation to Steven MacLean."
MacLean was ruled out for four months earlier this season after undergoing knee surgery.
St Johnstone continue their league campaign on Saturday with a trip to Kilmarnock.