The 21-year-old has been linked with a loan move in recent weeks, having made just one senior appearance for the club in the new campaign.
"I've spoken to Aidy and he wants to play left-back. That is his preferred position. He played a development squad match at left-back last week and he did really well," McDermott told the club's official website.
"Aidy has been here since he was nine years of age. He loves Leeds United, so does his dad and he wants to do well and he wants to give himself every chance at cracking it at Leeds United. If that is the case, I want to make sure that he gets every opportunity."
White was used as a winger by previous boss Neil Warnock.