The 19-year-old joined the League One side in August and soon made his debut in a 5-1 defeat to Leyton Orient. However, a thigh injury sustained in his next match, a Capital One Cup draw with Blackburn Rovers, has left him sidelined since.
"Reece is a great lad and he has been absolutely first class since the moment he came up here," caretaker manager Graham Kavanagh told Carlisle's official website.
"He settled quickly, he was getting games and he was just really unfortunate with the timing of his injury. I spoke to Manchester United about the situation and they have been really understanding with us."