The Liverpool goalkeeper picked up a knock during the tour of Asia and has been unable to train fully since.
At the time, Klopp said he expected the No. 1 to be ready for the Community Shield clash against Man City at the end of the month.
However, with just four days until the Reds come up against last season’s Premier League winners, Alisson is yet to return to full training and has been ruled out of Wednesday’s friendly against Red Bull Salzburg.
In the last few days he has taken part in light sessions away from the main goalkeeping group, and Klopp says he is closer to a return than any of Liverpool’s other injured players.
“Obviously Diogo [Jota] is not here, that’s clear, but from the boys who are injured but are here, Ali is the closest – definitely not for Salzburg but then we will see,” the Liverpool boss told Liverpoolfc.com.
“He is close to get back I think.
“Between now and five o’clock, it’s now 2pm and between now and 5pm I will wait for messages and see who can train this afternoon properly. Then the game tomorrow, we will see.
“But how all pre-season games are, they are important for us from a physical point of view, from a tactical point of view, from a growing point of view into ourselves again. Let’s see.”
Klopp has also revealed that none of his players will play 90 minutes against Salzburg, with the aim to push them through a full match in Saturday’s meeting with City instead.
With a shorter pre-season than usual this year, the coaching staff have been gradually increasing the players’ workload ahead of another hectic campaign.
“This pre-season we had to do slightly different and we wanted to do as well,” he continued.
“All of them will play the first time 90 minutes on Saturday, all of them. But that’s how we did the training.
“So the training sessions were intense to a specific point, that we could get fitter and fitter. The running sessions were extremely hard but we took care of them and now let’s see how we can grow in this season.
“For City, it’s not really different, they came back from America probably yesterday or something like that. So it will be a tough one.”