The 26-year-old had been on trial at Brunton Park and played for 90 minutes as the Cumbrians beat Rangers 3-2 in a behind-closed-doors friendly last week.
He will now stay with the League One side for a further month, during which time he will be eligible to play in competitive games.
Manager Greg Abbott told the club's official website: "It's an opportunity for Greg to have a look at us and for us to have a look at him.
"He played well against Rangers and this is a real opportunity for him to show us what he can do.
"He obviously now provides cover for the weekend and we can take it from there."
Fleming's previous clubs include Livingston, Gretna, Oldham, Galway, Chesterfield and Grimsby.