The England u-21 international has had a breakthrough season under Jurgen Klopp, making nine appearances across all competitions.
Quansah spent the second half of last season at League One side Bristol Rovers and was expected to move again temporarily before staying at Anfield.
Quansah has been praised by Klopp and other senior players at Liverpool for his performances but he has vowed to remain grounded and aim for the top.
He said: “No (his season has not exceeded expectations), because no-one has said ‘Jarell Quansah is one of the best players in the world’ yet.
“No-one’s said ‘he’s a top centre-back’. No-one’s said ‘he’s all this’, so until I’m regarded as at least one of the best players in the Premier League, then there is no reason for me to sit on what I’ve got and go ‘Oh yeah, I’ve played nine games for Liverpool’. That’s never been the end goal.
“I’ve always just wanted to be a winner. I’ve hated losing since I was young. I used to cry when I lost. When I lose, it still ruins my weekend and stuff like that.
“There is no point in setting goals that you think you could reach. You might as well just set a goal that maybe looks out of reach but you never know what you can get to.
“Surprised could be deemed an interesting word but I think he (Klopp) means I wasn’t expected to be there yet.
“I don’t think it was ‘It’s a surprise, I’ve trained with him before and he’s not a good player’. It was about how soon it’s come and how I’ve dealt with the adversity I have been thrown into.
“It’s not expected from being on loan at Bristol Rovers to going straight into the first team, and to do as well as I did. That’s where the surprise came from.”
