The company, which is joint owner of Meadowhall, is beginning a public consultation on the proposals.
The 30,660m2 Leisure Hall would add new restaurants and cafes, a replacement cinema, a gym and other leisure space.
The plans will go on display at Meadowhall early next month. A detailed planning application for the will be submitted in late summer. If the proposals are approved, British Land hopes to start on site during 2018 and open the Leisure Hall to visitors in late 2020/early 2021.