The 28-year-old suffered the injury earlier this month during his side's 3-3 draw with Hibernian in the Scottish Premiership.
"Lee could be out for anything up to a month," Motherwell boss Stuart McCall told the Evening Times. "The injury happened early on against Hibs and it says a lot about him that he played on for the rest of the match and produced some fantastic saves into the bargain.
"But unfortunately a scan revealed the extent of the injury and I really feel for Lee. He has not had his injury problems to seek of late but he has shown a fantastic attitude to get back."
Motherwell return to action on Wednesday night when they host Inverness Caledonian Thistle in the league.