The three councillors in whose ward the Regent Arcade stands say they are glad to see something being done with the site.
Despite only being just over a decade old, an application has been lodged with Swindon Borough Council's planning department to knock nearly all of it down.
The plan, put forward by Manchester-based MCR Property Group, would see the whole complex; supermarket unit, car park, cinema and other shops, bars and restaurants pulled down with only the unit hosting restaurant chain Nando's retained.
If approved the work could start as soon as January 30 - the date listed in the application notice posted on the sign board at the complex.
Eastcott Ward councillors were unanimous in their wish to see something done with the site even if that meant knocking the buildings down after only 11 years.
Councillor Paul Dixon said: "I welcome the proposal to start work on the site and hope that a full planning application will follow promptly and that the developer engaged positively with local residents with their plans as the work progresses.
"I want to see the site being redeveloped and brought back into use, it has been over 6 years since Morrisons closed.
"It is encouraging to see that a developer has seen the potential in Swindon and is prepared to invest in the town centre."
Councillor Imtiyaz Shaikh said he agreed with everything Cllr Dixon said.
Speaking solely in her capacity as a ward councillor, and not as the cabinet member for planning, Councillor Marina Strinkovsky, said: "It's a shame that the centre is so empty and we hope that a new use for it can be found as soon as possible."
Swindon Borough Council's planners have set themselves a target of making a decision on the plan by mid-January.