The controversial plan will involve removing remains from an old graveyard next to the former Swinton Unitarian Church. The bodies will be reinterred at Swinton Cemetery.
Once this has been completed Asda will build a 35,000 sq ft store above a 347-space ground floor car park.
An Asda spokesman said it would pay for the graves to be moved and would consult with relatives on the kind of memorials to be put in place.
The burial site includes three victims of the 1885 Clifton pit disaster – a mine explosion which claimed 178 lives.