Millennium Partners, which is redeveloping a key piece of Downtown Crossing, said Monday that Roche Bros.has begun construction on a 25,000- square-foot supermarket that is scheduled to open early next year.

The supermarket will be at the Millennium Tower/Burnham Building, and it will occupy much of the space that once housed local retail legend Filene’s Basement. Headquartered in Wellesley, Roche Bros. operates roughly 20 stores.

Millennium had big news in April when the Irish clothing chain Primark announced it would open its first US store in the historic Burnham Building.