American Apparel LLC is headed to the bankruptcy auction block next month with a deal that means a new home for its brands and hope for its Los Angeles-based manufacturing and distribution operations, but a big question mark for the fate of the retailer’s 110 stores.

A beleaguered manufacturer and retailer of young adult fashion, American Apparel returned to bankruptcy Monday after months of scrambling for money and a buyer, less than a year after an earlier court-approved chapter 11 turnaround ran off the rails, according...