Anthony Glossop, St Modwen chief executive, said the stadium would form part of a multifacility sports village.
He said: "The intention is to get a cluster of outdoor sports such as rugby and cricket that can share the facilities and car parking."
He added that the new site would not include retail but that there would be scope for a hotel.
The club is set to sell its current County Ground to the council. It will then work with St Modwen to redevelop the Shaw Tip site to the west of the town.
The vacated site, which is understood to be worth £7m, will be used for the development of new housing in what is a prime central location.
Swindon council leader Mike Bawden said there were several unresolved land issues but was excited by the plans.
He said: "I look forward to seeing the partnership's proposals take shape and, in the future, a detailed application for the planning committee to consider."
St Modwen has previously been in charge of developing Stoke City's new Britannia Stadium.