The 21-year-old, recently linked with a loan move to Sky Bet Championship rivals Barnsley, has made just one senior appearance for Leeds this season - against Chesterfield in the Capital One Cup.
White was employed as a winger by previous boss Neil Warnock, but has told McDermott he wants to stay at his hometown club and try to establish himself at left-back.
"I've spoken to Aidy and he wants to play left-back," McDermott told the club's official website.
"That is his preferred position. He played a development squad match at left-back last week and he did really well.
"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."