The Brazilian striker came on as a substitute in Wednesday's 2-0 win over Diego Simeone's side but lasted just 32 minutes before being forced off with the problem.
Giving a team news update to liverpoolfc.com ahead of the weekend's meeting with West Ham United, Klopp has now revealed that Firmino is set for yet another spell out of action with the issue.
Klopp said: "Bobby is not good news with a serious hamstring injury, really unfortunate. We don't know exactly how long he will be out but it will be not now after the international break directly. We have to work on that."
The Reds coach also ruled Curtis Jones and Joe Gomez out of the contest, adding: "Curtis is a completely different story [to Firmino], he was very unlucky in training – he got a finger that scratched the eye. I had a lot of injuries to players in my career, that was not involved. He is on the way back but we have to see how long exactly.
"He will be back after the international break, for sure, but in the international break we have to see when he can start doing proper stuff again. And Joey, hopefully after the international break as well."