Klopp had initially ruled the stopper out for “the next few weeks” after was forced off on the opening night of the Premier League, but the club held a more vague stance.
The latest update from goalkeeping coach John Achterberg was that the while Alisson had returned to outdoor work at Melwood, Liverpool “cannot look too far ahead” in terms of his recovery.
Speaking at the training ground on Friday ahead of the visit of Newcastle, Klopp provided an update on the Brazilian’s return, with late-October likely.
“Funnily enough we had lunch together again, so he looks good,” he said, recalling his last words on the 26-year-old.
“He’s improving, but we cannot put any pressure on it.
“It was a really serious injury in the calf, it’s now much better of course and that’s good for us, good for him. But we don’t know exactly.
“The target was always after the next international break, he could be ready [then] but we don’t know.
“How I said, there’s no pressure, and hopefully everything will be fine for the rest of the season. That’s the plan.”
There was more positive news on Naby Keita, however, with the Guinean on the mend after suffering a setback as he worked his way back into the fold at the start of the season.
Keita limped off during an open training session ahead of the UEFA Super Cup clash with Chelsea, but should be back in training next week.
“Naby is close, much closer than Ali, I think he could be training when we come back from Napoli next week,” Klopp explained, before providing a more general overview.
“Clyney and Paul Glatzel, they’re obviously far away still but doing big steps as well.
“I hope I didn’t miss anybody but it looks quite OK in the moment.”