Milner was absent for the 4-0 thrashing of Newcastle on Boxing Day, with Klopp forced to start Trent Alexander-Arnold days after his return from a foot injury.
The 20-year-old played the full 90 minutes at Anfield, and was in full training the following day in a major boost for the Reds ahead of two big clashes over the turn of the year.
“Milly is Milly: that means he’s again ahead of schedule,” he told reporters.
“He [only] has a little bit with the muscle, but he will not be available for tomorrow.
“That’s not possible, but hopefully the game after that he will be available again.
“Alberto trained for the first time really pain-free yesterday, so will be in contention. Dom Solanke has muscle problems, so will not be involved.
“Joe [Gomez] obviously not, Joel [Matip] not. Both are doing well, big steps, but will take some time.”
The absence of Milner should see Alexander-Arnold retained at right-back on Saturday, though Nathaniel Clyne is available as backup.
Andy Robertson has played the fourth-most minutes of any Liverpool player this season (2,054), but Klopp is unlikely to risk leaving his first-choice left-back out against either Arsenal or City despite Moreno’s return.
Gomez (leg), Matip (collarbone), Rhian Brewster (knee) and Alex Oxlade-Chamberlain (knee) are all long-term absentees, but Klopp confirmed on Friday that the latter could be back in full training as early as February.