The German takes his side to London on Friday, when Oxlade-Chamberlain will look to make a point against the club he left in August.
His recent performances have drawn praise from many, though Klopp was impressed from the start and suggested that he may be demanding more attacking output than was expected of the 24-year-old by Arsene Wenger.
"He did well from the first day," the Reds boss said. "If you are a big signing everyone expects you to perform immediately. It's difficult, you can't just play a little bit, you don't know all your team-mates - it's all different.
"He has become more and more a Liverpool player. That's how it is. He's really fit, you could see it at Bournemouth, there was not one second where you thought he looked tired.
"The biggest improvement and the biggest potential he has is being involved in goalscoring situations. He has it in him, but he maybe hasn't been asked to do that too much in the past.
"If you watch Arsenal in the past, their two decisive players were [Mesut] Ozil or [Alexis] Sanchez.
"Here we put it on more shoulders and if he's on the pitch he needs to be involved in situations like this as well."
2 - Only two of Alex Oxlade-Chamberlain's 20 goals for Arsenal came outside of London (away at Bournemouth & Nottm Forest). Venture. pic.twitter.com/STCJD7nfol— OptaJoe (@OptaJoe) August 31, 2017
Klopp is anticipating a stern test in the capital, with Arsene Wenger's side – who will be without the injured Olivier Giroud – sitting only one point behind them in the Premier League.
He said: "They are a complete side, able to do everything. Giroud can't play, but they have a lot of good, good footballers on the pitch. We have to defend them and use any spaces we can find. We go there to perform and get as much as we can.
"We are a team that is very offensive-orientated, but that doesn't mean you have to give any space away. The boys do the jobs now much more often."