The 19-year-old, a regular in the Under-21 squad at Old Trafford, will stay with the Cumbrians until January 1.
"We've watched Reece a number of times and he's one we think will add to the squad in a very positive way," manager Greg Abbott told the club website.
"He's keen and hungry to be involved and he's one we're really looking forward to working with.
"He's a good footballer and a hard worker and he really wants to be here. Hopefully this type of move will give him time to settle and make the type of impact we think he's capable of making."