Milner was withdrawn after 38 minutes of the Reds' fourth-round clash with Pep Guardiola's side at Etihad Stadium having pulled up.
Klopp explained during his post-match press conference: "Millie felt something in the hamstring. We still hope it is not an injury, but we will see. He felt something, but after 30 years in the business he said, 'I don't think it's a lot' so we will see."
Meanwhile, the manager explained how a virus kept Alexander-Arnold out of the fixture.
The full-back was set to be involved in the encounter until sickness ruled him out, with Alex Oxlade-Chamberlain also suffering from illness in the build-up to the game.
"We thought Trent was available but then he got ill, so that is what we have to wait for," Klopp said of Alexander-Arnold.
Elsewhere, Virgil van Dijk is set to return to the squad for Liverpool's Premier League fixture with Aston Villa on Boxing Day following World Cup duty.
"Virgil trained today [and] yesterday completely normal and should be ready for Monday, but we have to see who gets the virus until then," added the boss.