The 21-year-old, born and raised in the city, is one short of 100 games for the club he made his debut for in 2016.
He became the youngest player to start two consecutive Champions League finals when Liverpool beat Tottenham in June.
Alexander-Arnold also said he has never thought about leaving Liverpool and “can’t see that changing”.
Alexander-Arnold will reach a century of appearances if he plays against Aston Villa in the Premier League on Saturday, but the landmark is not his biggest ambition at the club.
The England international said:
“I am not shy in saying that [being captain] is a dream for me. Whether it comes true or not is not up to me; I don’t pick who is captain.
“But that is something I would love to do one day. Captaining Liverpool is something I have always dreamed of and it is something that motivates me.
“I have always been a Liverpool player, Liverpool has always been my home. I have never thought about changing clubs.
“I always thought Liverpool was destined to be my club. When I grew up, the dream was always to play for Liverpool. Now I am living the dream and I can’t see that changing.”
But he explained how a lottery draw when he was six years old set him on the path to where he is today.
“It was a normal day at school coming up to half term and there was a summer camp going on at the [Liverpool] academy,” he said. “They sent six or seven invites to my school. It was about encouraging young players to play football.
“Every child wanted to go, so names were picked out of a hat and mine came out. I went down on the Saturday with mum, nervous as anything – excited, couldn’t sleep.
“I was nervous to see if I was going to fit in, if I was any good or not. I went in the morning and it was within the first hours, one of the members of staff went over to my mum and said: ‘Can you start bringing him two or three times a week?’”