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".

Nerijus Gudelevicius would entice Lithuanians to the UK with the promise of paid work, delivering and collecting charity clothes bags.
The 34-year-old, of Westlea, Swindon, was found guilty at Swindon Crown Court of two charges of human trafficking.
He was cleared of a further four charges of slavery and released on conditional bail.
The court heard Gudelevicius would pay workers between Â£10 and Â£30 a week - but sometimes payments were made every fortnight or three weeks.
He would also charge them Â£40-a-week in rent, if they needed accommodation.
Gudelevicius, originally from Lithuania, is due to sentenced at Salisbury Crown Court on 19 June.
Following the judgement, acting Det Insp Rob Findlay said the verdict was a "positive result" following months of robust investigation work.
"The victims of human trafficking are often very vulnerable people who are used as simple commodities - often for the financial gain of the perpetrators," he said.
"Wiltshire Police will always treat human trafficking as a priority and vigorously investigate, along with our partners, such incidents.
"Community-based intelligence proves crucial in investigations such as this, and I would always urge anyone with information or concerns about potential victims of modern slavery to contact us."
Liverpool midfielder Adam Lallana is set to miss "a couple of months" because of a thigh injury, Reds' manager Jurgen Klopp has confirmed.
[NEXT_CONCEPT]
A man has been found guilty of human trafficking in what police have called a case of "modern slavery".