The 29-year-old sustained the injury during his side's Audi Cup final defeat by Atletico Madrid on Wednesday.
"I think we can pretty much rule him out of August and September," said Klopp. "This is certainly not news we would have wanted."
He could now miss England's World Cup qualifying games in September.
Lallana has played 33 times for England and Gareth Southgate's side face Malta on 1 September and then play Slovakia at Wembley on 4 September.
Liverpool suffered their first defeat of the pre-season campaign after losing 5-4 on penalties against Atletico in Munich after it had finished 1-1 at the end of 90 minutes.
The Reds will now prepare to face Athletic Bilbao in Dublin in their final friendly match on Saturday before their league campaign begins at Watford on 12 August.
Daniel Sturridge also picked up an injury in pre-season - the striker went off with a thigh problem on Tuesday after scoring in their 3-0 win over Bayern Munich.
He immediately pulled up and was substituted on 87 minutes, but Klopp said he thought it was "nothing serious".
Liverpool midfielder Adam Lallana is set to miss "a couple of months" because of a thigh injury, Reds' manager Jurgen Klopp has confirmed.