Alexander-Arnold has missed the Reds' last four fixtures, having gone off early in the 3-0 win over Aston Villa prior to the international break.
It was originally predicted that 24-year-old would only miss around two weeks with a minor hamstring strain. However, his recovery has been slower than expected.
Speaking ahead of his side's clash against Tottenham Hotspur on Saturday evening, Klopp gave an optimistic update on Alexander-Arnold, while also reporting on the statuses of Stefan Bajcetic, Thiago Alcantara and Conor Bradley.
He told reporters: "Trent trained yesterday normal and will train today normal, and then we have to make a decision.
"He will be in the squad, I'm not a doctor so cannot make that decision, but it looks like he can be in the squad tomorrow and play.
"Stefan had a little, little setback. The problem with Stefan is I can imagine everyone wants to see him play more – I want – but we have to be really careful. And now it's a little, little, little tiny calf issue. That's how it is when you are six months out.
"And we try everything to get him up to speed without pushing him. We make a very specific plan for him but in the end it's disappointing for him, of course, but for us as well that he got another one.
"So we have to take him out a little bit. He can work on pretty much everything else but not playing. That's it, I would say.
"Thiago still out, Conor still out. I think that's it."
Liverpool's trip to the Tottenham Hotspur Stadium promises to be their toughest test of the season so far.
Both Klopp and Ange Postecoglou's sides are unbeaten thus far in 2023/24, with Spurs currently fourth in the table, two points behind Liverpool.
Addressing the challenge that awaits Liverpool on Saturday evening, Klopp said: "Spurs is a tough test, we've not played an easy game yet, away at Newcastle, Chelsea really tough. All aways have been tough.
"Spurs doing well, Ange seems a top bloke, looking forward to meeting him. I think [fans] were all desperate for offensive football, he's doing it."