The 18-year-old, who has been tipped as a star of the future, will remain at Gresty Road until January 6 as he attempts to gain first-team appearance.
"Once I knew there was a possibility I could come to Crewe I wanted to come here. There were other interested clubs but I knew I would be joining a club who play football the right way," Bunn told the club's official website.
"I played about 18 games last season, mainly as a wide player, but I hope to play more in my favourite position up-front. I like to come deep and get the ball, turn, run at defenders and score goals.
"I have had a look around the stadium. It seems a big club. They used to be in the Championship and hopefully they can get back there."
Bunn has yet to make a first-team appearance for City.