Walcott, himself, was playing for Arsenal and England at 17 years of age.
Dowman made his Premier League debut in victory over Leeds on Saturday and Walcott told BBC Sport: "He's got ability where you just can't teach it, because he plays with that freedom.
"We talk about the love of football, football being boring at times to watch. Watching, he's not boring to watch.
"He's so positive every time he has it and that's what I love about these players that are just positive. It's so refreshing.
"He carries the ball, he's quicker with the ball. He's like Messi. I remember playing against Messi - when he was off the ball he's not quick. Maybe it's because I was quicker than most.
"When he had the ball he would just glide past people, glide past me with ease. (Dowman's) got that, he's got the stop-start."