The benches were suddenly removed from the centre in early June, sparking anger from local residents.
Residents have criticised the move, and even held sit down protests in the centre.
However, LCP said that the benches had been an "obstruction" to the pathways.
MP Rachel Reeves (Lab, Leeds West) and Bramely and Stanningley councillors, Kevin Ritchie, Caroline Gruen and Julie Heselwood raised the issue with the new owners, and have been in consultation with them about bringing back the benches.
A petition has now been started on Change.org calling on LCP to restore the benches.
More than 450 people have signed it.
The petition reads: "The seating at our local shopping parade has been removed by the new owners, who want to increase footfall to the centre and claim the seats were an obstruction.
"The lack of seating means many people are excluded from using the centre, for shopping and for vital social contact.
"Many people need to sit and rest while shopping, including some older people, people with disabilities or mobility problems, people with buggies, and staff in the shops taking breaks.
"Without any seating, older people have had to sit on bins in order to rest.
"Parents sit with children on the pavement. And others have simply stopped using the centre alltogether.
"Having a place to sit and see other people is important for local people's health and wellbeing.
"Having seating will increase trade and help retain tenants.
"Restoring the seats will bring people back to the centre, and revive the community's faith in the new management.
"We the undersigned ask LCP Group to reinstate the seating as soon as possible and create a more welcoming centre in Bramley that local people, customers, and visitors to the area can be proud of."
A spokesman for LCP, which manages Bramley Shopping Centre, said: “The benches were removed because they were causing an obstruction on the common area pathways.
“Any decisions made are done so with the best interests of our tenants and shoppers and we will consider if new outdoor seating arrangements can be made once we’ve had chance to assess the impact of the change.
"However, there is external seating outside Costa Coffee and retailers can request the same by applying to the landlord for its consideration.”
The Change.org petition can be viewed here.