The 18-year-old, who played in several pre-season friendlies while on trial at Turf Moor this summer, will join the Clarets' development squad.
Burnley's development squad manager Simon Weatherstone is looking forward to working with the promising youngster.
He said: "Kyle has done really well in pre-season and showed his quality. He is an attacking left-back who can put in a great delivery when he gets in to the final third and I know he was highly thought of at Stockport.
"He has trained with the first team alongside the rest of the development squad and was impressive.
"Stockport seems to be a good breeding ground for young talent at present.
"Danny Whitehead departed for West Ham earlier in the summer and now Kyle has signed terms here.
"Hopefully he can progress into the first team in years to come."