Liverpool are optimistic that Alisson will be in contention to face Manchester United at Old Trafford on Oct. 20.
The 26-year-old has been sidelined since coming off injured in the Premier League opener against Norwich City on Aug. 9, but stepped up his recovery at the club's Melwood training complex during the September international break.
Alisson was able to do footwork, catching and passing drills outside with his rehabilitation progressing as scheduled and is now pencilled in to return after the October international break which ends on Oct. 15.
The reigning European champions have a taxing challenge during that month, with a Champions League trip to Genk sandwiched between the visit to United and a game against Tottenham at Anfield.
Klopp is optimistic about selecting his first-choice goalkeeper for those fixtures, but he won't be rushing Alisson's recovery.
"We just had lunch together; Ali looks good and he is improving, but we cannot put any pressure on it," Klopp said during Friday's news conference.
"It was a really serious injury in the calf. It is now much better, of course, and that's good for us and good for him, but we don't know exactly [how long it will be].
"The target was always after the next international break [that] he could be ready, but we don't know.
"So, as I said, there is no pressure and hopefully everything will be fine for the rest of the season, that's the plan."
There will be no such wait for midfielder Naby Keita, who is due back in training next week following a hip problem.
The Guinea international suffered the setback ahead of the Super Cup final victory over Chelsea in Istanbul.
"Naby is close, much closer than Ali," Klopp, the Premier League Manager of the Month for August, said.
"I think he could be in training when we come back from Napoli next week."
Liverpool will line up against Carlo Ancelotti's side in Naples on Tuesday and the Italian manager has bemoaned the state of the dressing rooms at Stadio San Paolo following renovation work.
The home changing space remains an unfinished construction site, with the club posting a video of it on their Twitter feed on Thursday.
Queste le condizioni degli spogliatoi del Napoli dello stadio San Paolo, a 70 ore da #NapoliSampdoria, che hanno scatenato la reazione di @MrAncelotti contro Regione, Comune e Commissari. pic.twitter.com/YAROtscwIh— Official SSC Napoli (@sscnapoli) September 12, 2019
Klopp said "it would not be too cool" if Liverpool were exposed to the conditions that left Ancelotti "dismayed."
The visitors were informed their quarters are not affected by the building work, while UEFA said they are in contact with Napoli "to get all necessary assurances and guarantees that the dressing rooms and any other relevant facilities are ready and suitable for next week's Champions League match."
Klopp said: "I think it will be sorted. I heard different things, I saw pictures but I was not live streaming!
"I trust UEFA enough that they will sort so we have a shower and a space where we can change. I hope that will be organised.
"I don't want make it bigger than it is. If it looks like it looked on the pictures we saw, that would not be too cool, but as long as the pitch is there then we can play.
"The rest, we cannot change. We go there and accept the circumstances, and go out and play football."