The two parties have enjoyed progressive talks over the past 48 hours and the 21-year-old is due to line up in Carlisle's friendly against Hibernian on Sunday afternoon.
"James comes highly recommended and his loan spells with Plymouth and Peterborough will have given him some idea of what to expect from football at this level," Cumbrians boss Greg Abbott told the club's official website.
"The talks were really positive so this (friendly) will give all parties a chance to look at each other, and we can take it from there."
Chester himself said: "I was told about the enquiry as we set off (for Manchester United's friendly at St Johnstone) and I knew right away that it could give me the chance to play some first-team football.
"I spoke to the staff at Old Trafford and they told me good things, so I had no hesitation in coming up here. I'm looking forward to getting on with it now."