The county council is currently looking for an external organisation to potentially deliver library services on its behalf.
The successful bidder will be responsible for providing the new-look service approved by the council’s Executive earlier this year.
Changes include Gainsborough library being open will be open for at least 45 hours per week, over six days, closing at 6pm on at least one day.
All these libraries will offer free access to computers and the internet, along with other services like printing.
The authority has received initial applications from Compass Point Business Services, Greenwich Leisure Limited, Leisure in the Community, Lincs Inspire, Vision and Vivacity.
If an organisation is not found, the council will continue to deliver all of this with its own staff.
Coun Nick Worth, Executive Member for Libraries, said: “Although we’re facing a second legal challenge, we believe we took the right decision and took it lawfully, so we intend to push on with our plans.”
“The way in which our libraries are used has changed and it’s important they adapt with the times.”
“In addition, we can’t ignore the fact that by the end of this decade the council’s overall budget will have been almost halved.”
“We believe it’s only fair that the library service plays its part in finding the necessary savings.”
“However, under our proposals, we can do this in a way that both saves money and creates a service that’s fit for the 21st century.”
An initial evaluation will now take place, with short-listed organisations then asked to submit a detailed bid.
A final decision is expected to be made at the end of the year.