Alisson is walking without crutches but Liverpool boss Jurgen Klopp remains determined not to rush the goalkeeper back into action.
The Brazil international suffered a calf injury during the opening fixture of the Premier League season against Norwich City a fortnight ago.
Deputy keeper Adrian came on as a substitute and started the penalty shoot-out win over Chelsea in the UEFA Super Cup and the 2-1 league victory at Southampton last weekend.
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The former West Ham man saved a penalty from Tammy Abraham in the Super Cup shootout but committed an error to gift Southampton a goal and put Liverpool's win under threat.
Alisson has resumed fitness work in the gym, but Klopp appears ready to be patient over his return.
"Not that close," Klopp said of the 26-year-old on Friday when asked how near he was to playing again.
"I had lunch with him today so at least he can eat! He looks good, good shape, good mood, but it will take time. I don't know how long exactly.
"[There is] No date really in my mind, or nobody told me a date. We didn't ask [the medical department]; we want to give him the time he needs.
"At the beginning, he was on crutches, which is normal with the injury he had. They are not there now so he walks normally. He can really train now, [which is] completely different to a week ago.
"I think now it will go quickly but I don't know how quick."
Liverpool also remains without Naby Keita, who will miss Saturday's match against Arsenal due to a muscle strain.
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The Reds won 5-1 when the Gunners visited Anfield last December but, with the teams level on six points after two games this term, Klopp is braced for a tighter contest.
"I wouldn't expect [a high-scoring match], to be honest," he said.
"Both teams are probably offensively orientated, at least. Arsenal will probably change at least two key positions. Probably [Nicolas] Pepe will start, which does not make them weaker, and [Granit] Xhaka.
"It happened a lot to us in the last few months: we analyse a team and then they play completely different against us.
"I am not saying Arsenal will, but it makes life tougher, especially in the early stage of the season."
He added: "If you look at how they set up, they have good footballers in each position and real speed up front. Wow, real speed.
"That makes it tricky to defend. You have to be brave, play your own football, try to keep possession, be direct.
"There are a lot of things to consider in this game."
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