The 20-year-old will join his new team-mates for pre-season training next week after the Sky Blues agreed a deal with Sir Alex Ferguson.
Manager Andy Thorn said Brown, who has previously enjoyed loan spells with Oldham Athletic, Doncaster Rovers and Bradford City, would play a big part in the coming season.
He said: "This is another really good signing for us. Reece is a massive talent and we are delighted to be able to bring him here for the season.
“At 6ft 3ins, he’s big and athletic and can play at the back and in midfield which is a big plus for us. The fact that we looked after Ollie Norwood so well last season has stood us in good stead with Manchester United and we are grateful that Sir Alex Ferguson and his staff have put their faith in us to do exactly the same with Reece."