"The Argentinian defender experienced the setback in our Premier League game against Arsenal earlier this month," the club said in a statement.
"Martinez bravely continued playing for the following two games against Brighton & Hove Albion and Bayern Munich.
"But it has now been determined that he needs a period on the sidelines for recovery and rehabilitation. Assessment is ongoing to decide next steps."
Sergio Reguilon will also be absent from Saturday's Premier League game against Crystal Palace with a minor injury.
Ten Hag said that midfielders Scott McTominay and Christian Eriksen would return to the side having missed the League Cup victory through illness.
The Dutchman also expected a tougher encounter against Palace this time at Old Trafford.
"Definitely, both sides will be different but we got belief from it (Tuesday's win). I know the opponent is looking for revenge and that will be their view," he said.