Milner has impressed during his loan stint with The Robins after arriving at The County Ground at the start of September in search of first-team experience.
Robins boss Andy King is prepared to ask Reid if he can keep Milner with the Second Division outfit for a further month.
But Reid has warned King he will be wasting his breath as the Whites boss wants the promising 17-year-old back at his disposal.
"Andy King hasn't asked me about extending the loan and I've spoken to him over the past couple of days," Reid told Leeds' official website.
"But I only wanted James out on loan for a month to get a bit of experience so he'll be back with us at the end of the month.
"I know Andy very well and he will help James a lot, so it can only be good for his footballing education."
King has been delighted with Milner's impact at the club and will not be deterred from attempting to persuade Reid to allow him to keep the England youth international for a longer period.
"The boy is a terrific talent and everyone has been able to see the skills he has and I have no doubt he will go on to perform in The Premiership," said King.
"Unfortunately, he is not my player and I think, when Peter Reid says a month, he means a month. However that will not stop me asking the question and you never know. What I do know is the boy is enjoying his football."
