The leader of Lincolnshire County Council has said urgent talks will be held with a not-for-profit firm in a bid to save its library services.
It comes after the High Court ruled that the authority's plans to re-design the service were legally flawed.
Councillor Martin Hill questioned how Greenwich Leisure Limited (GLL) could save the authority £2m, without cutting jobs and "still retain the service".
The county council wants 30 of its 45 libraries to be run by volunteers.
It offered to allow communities to run them or they could have faced closure. The authority would have continued running the remaining 15.
Campaign group Save Lincolnshire Libraries, which won the legal battle, had collected a petition containing 23,000 signatures opposing the council's plans.
Mr Justice Collins ruled the public consultation had been flawed and the council had failed to deal properly with an alternative proposal.
After the ruling, Mr Hill said he was "sorry" about the way the authority had handled the situation and would discuss his concerns with GLL.
"The issue we have with Greenwich Leisure was that they were claiming they could save this money without losing any staff and any libraries," he said.
"Frankly I don't know how it's possible to save £2m without using fewer staff."
Mr Hill added that GLL has been contacted and were looking for a way to "move forward".
Diana Edmonds, from GLL, a charitable social enterprise, which had offered to run the libraries, said: "We have opportunities to make tax savings that the local authorities don't have.
"We don't expect people to do full-time jobs if they are not being paid for it."
Ms Edmonds added that its proposal would not lead to the closure of mobile library services in Lincolnshire.
Speaking on BBC Radio Lincolnshire on Thursday, Prime Minister David Cameron said: "The idea of getting volunteers to help with the running of libraries, that has worked in other parts of the country, and obviously we have to make sure we provide services efficiently, effectively, and we encourage volunteers and other organisations to help with the provision of some of these services."