The plans follow a £60m refurbishment of the shopping centre, which is due to be completed by the end of 2017.
The refurbishment hopes to continue the shopping centre's successful leasing strategy, which has seen brands such as Ghost, Urban Decay, Michael Kors and Diesel open stores in the centre in recent months. It will also offer dining and entertainment options, which are not currently available at Meadowhall.
In addition, the extension will provide internal and external spaces for events and community use, including new restaurants, a state-of-the-art cinema (to replace the existing cinema), a new café court, gym and additional leisure space to offer new experiences and activities.
Charles Maudsley, head of retail and leisure for British Land, the company which has submitted the application, said: "The exciting £60m refurbishment completing next year, the continued leasing success and Leisure Hall plans combined reinforce our commitment to ensuring Meadowhall remains an outstanding place for modern consumer lifestyles."
Last month, car brand Citroën entered into 15 month partnership with Stratford Westfield and will stage a series of events and customer experiences. On 29 October, LEGO unveiled their new 'Yellow Submarine' set at shopping centre Liverpool One.
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