The 21-year-old striker has been an unused substitute on five occasions for the Whites this season but has yet to make his senior debut.
"I would enjoy going out on loan if there's no chance for me of getting first-team football here," Erwin told Leeds' official website. "After getting first-team football last year, it's a lot better than playing Under-21 football.
"Hopefully I can get the chance to prove my point. The main thing for me is just to develop. I'm still just 21."
Erwin is expected to go straight into David Flitcroft's squad for Bury's home league game against Wigan on Saturday.
He signed a three-year contract when joining Leeds from Motherwell in June, shortly after helping the Scottish side preserve their top-flight status by beating Rangers in the play-offs.