The 29-year-old's injury comes at a time when the champions are already missing defenders Virgil van Dijk and Joe Gomez with long-term issues.
"The normal stuff with these kind of injuries ... (is) around about three weeks," Klopp told a news conference following Wednesday's 0-0 draw with Newcastle United.
Liverpool top the table with 33 points from 16 games and midfielder James Milner said he was pleased the side are setting the pace despite some lacklustre displays.
"I think it's a positive but it seems to be at this moment in time that nobody wants to take it by the scruff of the neck. We've dropped points, everyone seems to be dropping points," Milner said.
Liverpool will have to host home games behind closed doors again after the Liverpool City Region was placed into Tier 3 of Britain's COVID-19 restrictions on Wednesday.
"It's important that we need to kick on now and try to get a few three-pointers. But the world is in a strange place at the moment and it's the same in football," Milner added.
"Everybody's lives are upside down, it's tough mentally for everybody and it's the same in football."
Liverpool travel to Southampton for their first game of 2021 on Monday.
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