Chester is with Carlisle until the end of the year.
"James has been absolutely superb," the United boss told the News & Star.
"If you ask me now whether I would like to take him for longer, then yes I would.
"We just need to let him settle a bit longer. James doesn't say too much but I have spoken to his reserve team coach [Warren Joyce] and he is absolutely loving it here. "We will see what he says in the next couple of weeks. If it is positive, we will speak to Manchester United about their thoughts and see where it goes from there."