The Scot was forced to go under the knife after dislocating the joint while representing his country during the October international break.
He has been sidelined ever since but, according to Klopp, remains on track for a comeback in the New Year.
In fact, Robertson is starting to get frustrated that he is not able to get involved in training despite now only feeling minor discomfort in his shoulder.
Klopp: “Andy, I sat with him yesterday and he is doing pretty well.
“The problem is that the shoulder is not there, the rest of the body’s fine, he needs to be patient a little bit longer.
“He can do all the physical work already, which annoys him a lot because he is not allowed to do ball training, a little bit of passing maybe but not properly.
“But when he is allowed to go shoulder-wise, I think it’s a short time until he will be back. We all hope it will be in January, but I don’t know.”
As for Liverpool’s other injuries, Klopp confirmed Ryan Gravenberch is “fine” for the visit of Arsenal, and while Alexis Mac Allister and Diogo Jota are “getting closer,” they will “not be ready” for Saturday’s match.