The Brazilian has been sidelined from the opening night of the Premier League, with a calf injury sustained against Norwich keeping him out for almost two months.
But he has made strong progress in recent weeks, and after a series of individual sessions with John Achterberg at Melwood, rejoined the first-team squad on Thursday.
This could make him available for the visit of Leicester on Saturday afternoon, and though it is unlikely Klopp will rush him back—aided by the form of Adrian—the manager told reporters on Friday he is yet to decide.
“With Alisson it’s still maybe, maybe not,” he explained.
“Yesterday he trained with the team [for the first time], if you know goalie sessions they are very often more intense than the proper football sessions; [he] looked really good.
“So we will see, I have to talk to different people before I make a decision about that.”
Joel Matip will, unfortunately, miss out, with a knock keeping him sidelined until after the international break.