The 19-year-old will remain at Gresty Road until January 6 and joins up with his new team-mates ahead of Tuesday night's League One clash at Scunthorpe, where boss Steve Davis is hoping to receive clearance for the new recruit to feature.
Bunn is yet to make his first-team debut for the Premier League champions, but has a prolific scoring record for City's youth and reserve teams and enjoyed spells with Rochdale, Preston and Oldham last season.
"Once I knew there was a possibility I could come to Crewe I wanted to come here," Bunn told the Railwaymen's official website.
"There were other interested clubs but I knew I would be joining a club who play football the right way.
"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."