The 22-year-old has been a regular for the Burnley under-23s so far this season.
He played 29 games for League One team Burton Albion on loan last term.
Nartey was released by Chelsea in the summer and has found his next club.
"It's a relief. After the whole Covid situation I was just happy getting back into the swing of it," said Nartey to the club website after signing.
"Burnley let me come originally, just to get fit, which I was always grateful for, regardless. Now I've signed here is a great feeling for me.
"Everyone here made me feel like I had been here for years. I felt comfortable settling down and they have been a huge help in making me feel comfortable up here.
"You reach an age at Chelsea where you want to break through.
"The loan last year helped me a lot but coming here with that family feeling I feel like I have a direction of where I'm going and progression. I think it's a new chapter in my life."