The move was confirmed at a Southwark Council cabinet meeting on Wednesday.
Peter John, leader of Southwark Council, said the proposal would "deliver first-rate communications services for our two boroughs by means of a social enterprise company".
A spokesman said the council had not made a final decision over the legal form of the company, and it might be jointly owned by the two boroughs. There is no clear timescale for the process.
Celia Richardson, director of communications at the Social Enterprise Coalition, said the councils’ plans seemed confused.
"A business owned by a public sector body can’t be a social enterprise," she said.
"As the public sector changes, we are going to see lots of attempts to create arm’s-length bodies and management buyouts, and they will also want to use the social enterprise badge.
"We’re worried that we will see a lot of confusion among public sector organisations about what a social enterprise is."
Richardson said that if a social enterprise was to emerge from a public sector organisation, a cultural change was needed, with staff involved in the process rather than having it dictated to them by management.