The defender had to be substituted during their 2-1 loss to Dundee on Wednesday after developing a stomach muscle problem.
The club's manager, Danny Lennon, told the Daily Record: "It isn't looking good for Lee. He's struggling for the cup final which is a major blow. Not just for us but for the boy as well.
"It's possible he might need surgery so it could be weeks leading to months. And we're not prepared to take any risks with him.
"He'll be bitterly disappointed but knowing Lee's character and the strength of the man, I know he'll become a supporter and get behind us."
St Mirren will play Hearts in the final next Sunday.