The 34-year-old lasted just nine minutes of Sunday's FA Cup third-round win over Everton.
Speaking after the game, Klopp said: "Two players [Milner and Joe Gomez] started the last game and one of them got injured, so that's the situation we are in.
"That's why a lot of managers make a lot of changes; it's nothing to do with the cup, it's the timing.
"[Milner] has pain, that says it's serious. It's [his] muscle."