The 24-year-old completed a £40m move to Liverpool from their title rivals Arsenal on transfer deadline day this summer.
Oxlade-Chamberlain snubbed the chance to move to Premier League champions Chelsea despite passing a medical at the west London club.
The England international turned down the chance to sign a new contract with Arsenal in favour of a move to Anfield.
One of Oxlade-Chamberlain’s motivations behind his move to Liverpool was the prospect of securing a regular starting spot in a central midfield role.
However, the summer signing didn’t make his first Liverpool start until a 3-0 win against Slovenian side Maribor in the Champions League.
The England midfielder scored his first goal against their Group F opponents before he retained his place in the Liverpool team last weekend.
Oxlade-Chamberlain scored Liverpool’s second goal in a 4-1 victory over West Ham United at the London Stadium on his first Premier League start for the Reds.
The Liverpool summer signing sent a message to Klopp about his plans to secure a new starting spot.
“It’s never easy coming into a new team with as much quality as we’ve got here. It takes time to settle in and learn a new formation and style of play,” Oxlade-Chamberlain told Liverpool’s website.
“I’ve had to be patient; I’d have liked to have played as much as I can, that’s natural. The only thing I can do when I get the chance is try to impress and help the team to win, with goals and creating chances.
“I’m going to keep trying to do that and keep pushing for starting places, just as everyone else will. That’s the best thing – competition for places.
“The most important thing is that we’re winning. I’m happy with the result and obviously it was nice to start as well.”
Liverpool finished in fourth place in the Premier League table to secure a return to the Champions League.
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