Robertson joined Liverpool from Hull City in the summer transfer window as Klopp moved quickly to bolster his defensive options at Anfield.
However, the 23-year-old has found his first-team opportunities to be limited so far this season, having only made two appearances in the Premier League this term.
The Scot will be hoping to be involved when Liverpool host their bitter rivals Manchester United at Anfield on Saturday lunchtime.
And former Reds star Aldridge believes that Robertson deserves a chance to impress in the big game on Merseyside.
Speaking to the Liverpool Echo, Aldridge said: “I think the team pretty much picks itself for the weekend – but there is one change I would really like to see.
“I would love Andy Robertson to start at left-back.
“I like the look of this lad, I like his left foot and the balance he gives going forward. He also looks a better defender than Alberto Moreno, from the little bits I’ve seen.
“When he’s started for us, he’s had an impact. I think United will try to lock down the middle of the pitch, so we’ll be pushed out to the flanks.
“That’s where Robertson, with those crosses he can whip in between goalkeeper and defender, can be a real asset.”
Liverpool head into the game looking to get back to winning ways after the disappointment of their 1-1 draw with Newcastle United before the international break.
The Reds are currently in seventh place in the Premier League table and seven points behind leaders Manchester City as they look to try and claw their way back into the title race.
Manchester United, meanwhile, have been in fine form in recent weeks and are second in the table, having only dropped points on one occasion so far this season.
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