The proposed Leisure Hall at the South Yorkshire centre would create up to 1,400 jobs.
The 330,000 sq ft (27,800 sq m) facility would include a new cinema, restaurants, gym and new community spaces.
Joint owner British Land said the extension would "continue Meadowhall's evolution".
The plans, part of a five-year vision to extend the leisure offer at Meadowhall, include:
Around 540 jobs would be created during construction, in addition to 1,400 new jobs once the leisure hall is complete.
Sheffield City Council is expected to make a decision on the application in spring.
Darren Pearce, Meadowhall centre director and president of Sheffield Chamber of Commerce, said the extension would be a "major milestone" for the shopping centre.
Richard Wright, Sheffield Chamber of Commerce executive director, said it would be a "jewel in the Sheffield City Region's crown".
He added: "The Leisure Hall will retain spend in Sheffield city region by ensuring residents do not need to travel out of the area to fulfil their shopping and leisure needs.
"We welcome British Land's exciting plans and lend our wholehearted support.
"They will reinvigorate Meadowhall and serve as a catalyst for further investment in the region."
In Sheffield city centre, the current demolition of the Grosvenor House Hotel is under way.
It will be replaced by a new £90m HSBC building in the new year as part of the  Sheffield Retail Quarter.
A new city centre Primark store recently opened and a nine-screen cinema is due to open in the near future.

Wydad were 1-0 winners over 10-man Zanaco of Zambia to move top of the group, with record eight-time champions Al Ahly beating Cameroon's Coton Sport 3-1.
That put Ahly level on points with Zanaco, and with identical head-to-heads, Ahly went through, above the Zambians, on goal difference.
Zanaco knew a point in Casablanca would have been enough to put them into the last eight, but they were thwarted by a 68th-minute goal from Wydad's Achraf Bencharki in front of the 80,000-capacity stadium.
Bencharki's goal came nine minutes after Zanaco defender Taonga Bwenbya was sent off having been yellow-carded a second time.
It was the first Champions League loss suffered by Zanaco this season and only the third goal they had conceded after four wins and five draws in qualifying and group matches.
Al Ahly, who began their match against Coton Sport knowing only a win would take them through to the knockout stage for the first time since 2013, suffered an early setback.
They fell behind at the Borg El Arab Stadium in Alexandria after 12 minutes when veteran defender Ahmed Fathy conceded an own-goal.
Amr Gamal came to Ahly's rescue with an equaliser and he also scored the third Ahly goal after Abdallah El Said had given the Cairo club a 2-1 half-time lead.
It means four North African clubs have now reach the Champions League quarter-finals, with Esperance and Etoile Sahel of Tunisia already through.
There will also be at least two quarter-finalists from southern Africa - defending champions Mamelodi Sundowns of South Africa and Ferroviario Beira of Mozambique who went through after Al Hilal and Al Merreikh were disqualified following Sudan's global Fifa ban.
Detailed plans have been submitted for a £300m extension of Sheffield's Meadowhall shopping centre.
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Al Ahly of Egypt and Wydad Casablanca of Morocco secured their place in the African Champions League quarter-finals on Saturday as North African clubs prevailed in the final Group D matches.