A series of drop-in sessions, as part of a three-month review into the service, will start next week.
Essentially Powys County Council (PCC) cabinet would like volunteers, communities, town and community councils, partner agencies or charities to step forward and take them over.
And it is expected that the funding cut will affect staffing at the libraries.
At the full council meeting of PCC on January 24, questions were asked about Hay-on-Wye library.
Hay is the town famous worldwide for its book festival.
At the meeting county councillor Gareth Ratcliffe, Liberal Democrat for Hay-on-Wye, addressed concerns about Hay library.
He said: “Last year £1.2 million was spent on running library services, and there is a proposal to cut £200,000 or 16.7 per cent and this would be spread across 10 libraries.
“Have the bigger ones been asked to make savings?”
Councillor Rachel Powell answered: “We have to be smart as we move forward with sustainable methods.
“Currently the focus will be on the 10 smaller branches and how we can support them to be as self-sufficient as possible.”
Hay-on-Wye Library Supporters (HOWLS) have already said that they fear Hay would be in the firing line again.
They have said they will be meeting soon to discuss the situation.
Llanfair Caereinion Library, based at The Institute, is also under threat.
The town's council had already given everything to keep it open as a result of a similar review of services in 2016.
In a statement online they said: “The recent attack on our library was staved off by the town council taking over the running costs.
“The Institute halved the rent to aid this. There are volunteers manning it except for one member of staff.
“All heating, electricity etc are included in the rent so the cost to PCC is nominal.”
Councillor Rachel Powell, Portfolio Holder for Young People and Culture, said: “Within the previous administration, the service worked with those communities, various partners and managed to co-locate several of the libraries.
“Local members and town and community councils contributed significantly, as well as outside bodies and volunteers.
“Each library was unique in itself and successfully, libraries remained for residents, as well as providing efficiencies for the council.”
Councillor Powell added that she hoped the drop-in sessions would see ideas and solutions emerge to try and keep the libraries and mobile libraries open.
The drop in dates are as follows:
Monday, February 4: Hay-on-Wye – 10am to 12pm, and Talgarth – 1pm to 3pm.
Tuesday, February 5: Llanfair Caereinion – 10 am to 12pm, and Llanfyllin – 1pm to 3pm.
Wednesday, February 6: Knighton – 10am to 12pm, and Presteigne – 1pm to 3pm.
Thursday, February 7: Llanwrtyd Wells – 10am to 12pm, and Builth Wells – 1pm to 3pm.
Friday, February 8: Llanidloes 9.30am to 11.30am, and Rhayader 12.30pm to 2.30pm.
People can also take part in the review online until April 28 by visiting www.powys.gov.uk/haveyoursay
By Elgan Hearn, Local Democracy Reporter