Alisson suffered a calf injury while taking a goal kick during last Friday's opening 4-1 Premier League win over Norwich City.
It means close-season recruit Adrian will be between the posts when Liverpool contest the UEFA Super Cup against Chelsea in Istanbul on Wednesday.
Although Klopp was unable to give a specific return date for Alisson, he suggested reports of a six-week absence could be wide of the mark.
"[It's] Not too cool. It's a calf injury which takes him obviously out for a while," he told Liverpool's official website.
"I don't want to make now [an] exact prognosis on when he will be back, but it will not be tomorrow for sure, so now then we have to see. It takes a while, it takes a couple [of] weeks for sure, and we have to see.
"I saw already [people saying] six weeks and stuff like this, but Ali was not a lot injured in his career, so I would like to wait a little bit to see how he develops in that process now. But he is not in for the next few weeks."
Jürgen Klopp has confirmed @Alissonbecker will be sidelined for ‘the next few weeks’ due to the calf injury he sustained on Friday.— Liverpool FC (@LFC) August 12, 2019