The England international spent six years of his career at the North London outfit, but failed to establish himself as a first-team regular.
The 24-year-old closed the door on his Arsenal career on transfer deadline day, but he is yet to earn a starting spot in Klopp's team having made three substitute appearances.
The Southampton academy product will, however, get his chance on Tuesday when he will be named in the first XI for the Reds' EFL Cup tie against Leicester City.
"It would be strange for me to say it after two-and-a-half weeks when Arsene Wenger said this after working with him for six years, but a new start is very often a kind of relief, that's how it is," The Guardian quotes Klopp as saying.
"You are in a situation with your old club - and I don't say this about Alex specifically - and you accept it and it is difficult to make the next step. I thought it made complete sense for Alex to change club and to come here. So far I can't see that [he lacks self-belief].
"He has settled in perfectly. He knows the situation. He played all the games for Arsenal from the beginning and now he comes here and is on the bench so it looks 'Oh, not the best decision' but it is long-term thing. We want to use him and to prepare him also. It is all good from this side at the moment."
Liverpool currently sit eighth in the Premier League table.