The 32-year-old, who has 19 goals, has been nursing the injury since the start of the season and sometimes needs injections to play. Bayern Munich's Arjen Robben, also currently injured, is second on 17 goals.
Tests done in Basel, Switzerland on Monday revealed the extent of the injury and the need for surgery, Frankfurt said.
"The absence of Alex is a blow for us because we had hoped he could help us with his goals in the last games of the season," said coach Thomas Schaaf.
"We wish him all the best for the operation and to return healthy as quickly as possible."
Eintracht are in eighth place in the Bundesliga and have an outside chance of a European spot with six matches left.
(Reporting by Karolos Grohmann; editing by Martyn Herman)
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