Brazil international Alisson limped off in the first half of Liverpool's Premier League opener at Anfield on Friday.
"Not too cool. It's a calf injury which takes him obviously out for a while," Klopp told Liverpool's website.
"I don't want to make now an exact prognosis on when he will be back but it will not be Wednesday for sure, so now then we have to see. It takes a while, it takes a couple weeks for sure, and we have to see.
"I saw already like '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."
Alisson, 26, was replaced by experienced Spaniard Adrian against Norwich. The former West Ham keeper, who joined on a free transfer last week, is likely to keep his place for Liverpool's European Super Cup date with Chelsea in Istanbul on Wednesday.
Third-choice Liverpool keeper Caoimhin Kelleher is not fully fit after a broken wrist, so the club are set to sign 35-year-old former Middlesbrough player Andy Lonergan on a free transfer.
Lonergan was part of Liverpool's pre-season tour as cover for their other keepers.
Asked if Lonergan will be on the bench on Wednesday, Klopp said: "I think so. I didn't speak to (goalkeeping coach) John Achterberg yet about that. We spoke a lot about the goalie situation apart from what we have to do on Wednesday!
"We will see, but we can have obviously a lot of players on the bench there, which is good and I think Andy will be one of them.
"He was in with us, he's a brilliant character, a really nice lad and helped us a lot in the pre-season, and now we decided to work together with him."
Alisson's loss for an extended period would be a big blow to the club he joined from Roma last year.
But Liverpool defender Trent Alexander-Arnold, speaking after the Norwich game, insisted Adrian will prove a quality deputRead More