Shropshire Council has started consultation into plans to move Ellesmere Library from its base in Victoria Street to the nearby Meres Day Centre.
The council has said that a second option – to keep the library under Shropshire Council management – would have to see opening hours cut by more than half to make the same savings of £8,830 per year.
A third option is to put forward an alternative for the council to consider.
Shropshire Council said in a statement that having looked at a number of options, its preferred solution would be to move it to the nearby day centre.
The move would see library and customer service point services delivered there and there would be "appropriate alterations" to the space.
It added that the move would see the library's opening hours maintained or extended and an opportunity to "create a community hub" in the day centre.
It said: "This option would potentially secure the future of the library and Customer Service Point, providing the opportunity to develop a vibrant community hub offering a diverse range of important community services and support to potentially vulnerable people for the residents of Ellesmere and the surrounding area.
"Whilst this is our preferred option, a second proposal is included in this consultation document for the ongoing management by us of the library and Customer Service Point within its existing location at Fullwood House.
"To make the same savings as the preferred option the library opening hours would reduce from 20.5 hours to ten hours per week."
The council announced earlier this year that it was looking to save £1.3 million by finding alternative management for some of its library services including Ellesmere, Shawbury, Broseley, Shifnal and Church Stretton.
And while many users have said they hope to see the library stay were it is, the council warned that "there is a recognition that further financial pressure may come on library and customer service point services" and that an on-going review may be necessary.
The statement said: "We are now seeking responses to the proposals outlined.
"We want to stress that we have an open mind and no formal decision has yet been made and that there is an opportunity to make alternative proposals that can be delivered for the same, or a lower budget than the proposals listed detailed.
"The results of the consultation will be collated, analysed and considered, then used to inform a decision on the future delivery of the library and customer service point services in Ellesmere."
Steve Charmley, Shropshire Council's Cabinet member for business and culture, said: "Libraries are an integral part of the community and this is why we have been working extremely hard with our partners to ensure we can find sustainable solutions to enable libraries to continue to thrive for future generations to come.
"I hope that this consultation will attract a high level of response to help us find the best way to maintain the library in Ellesmere, within the budgets available.
"We very much look forward to hearing people's views on these proposals, particularly any alternative options that people may want to put forward."
Comments can be submitted online by going to shropshire.gov.uk/get-involved and searching for Ellesmere Library.
Residents can also pick up a questionnaire along with the proposals and frequently-asked questions at Ellesmere Library, Ellesmere Town Council offices, Ellesmere Post Office, and Ellesmere Newsagents.