He has also suggested that new head coach Thomas Tuchel consider him for an England spot.
The Gunners legend, who won three league titles with Arsenal, has watched Lewis-Skelly develop in the club's Hale End academy.
“I’ve got to say, his development has been sensational,” Winterburn told The Mail.
“He's now asking the question of Mikel Arteta: dare you leave me out the team? Because his performances have been so good.
“I’d like to see him be more aggressive in that full-back role defending, particularly shutting the opposition team down, pushing them into areas where he wants them to go or the team want them to go or what they've been working on in training.
“On the ball, he protects the ball really well, you can see that midfield role, he's a good mover with the ball, that's never been a doubt in my mind.
“I’ve watched Myles come through the youth system as I was doing some co-commentating for Arsenal. He was a midfield player initially but under Mikel Arteta he has been converted to that left-back role.
“I think he's even become even more physically stronger. The Premier League isn't physical anymore, but the physical side to protect the ball, he's easily capable of doing that
“I can just see him progressing and getting stronger and stronger in terms of that position because, let's be fair, he hasn't played in that position much.
“He should just become so much stronger in that position, in terms of his understanding of the role, almost dictating to your team-mates where you want them to be, the areas you want to push the opposition into, but you can see that he doesn't play with any fear, as young players shouldn’t.”