Denbighshire County Council Leader Jason McClellan said: "I do really feel for the staff. It's a difficult time."
A North Wales council leader who warned that Denbighshire Council "could face bankruptcy" has told ITV Cymru Wales he is determined "it will not come to that".
The local authority is facing a shortfall of between £20 million and £26 million for the next financial year and the cost may be pushed onto local residents with a possible council tax increase.
It went up by 3% this year.
Labour Leader of the council Jason McClellan acknowledges the local authority is facing what he described as "very tough times, very challenging times, I would use the word unprecedented".
"Nothing is off the table," he said.
"For example, at the moment we are looking at a consultation on our libraries. I didn't come into politics to shut libraries, libraries are an important part of our community. We are looking at perhaps closing one library one day, whilst another neighbouring library is open.
"That will save us a lot of money, but it will impact on our services."
Councillor McLellan, raised concerns about the state of the council's finances in a letter to councillors, he warned: “The main priority for cabinet at this point is to stop the council from going bankrupt."
He told ITV Cymru Wales: "Denbighshire isn't alone in this. Councils across England and Wales are facing exactly the same decisions and the same predicament
"I do really feel for the staff. It's a difficult time. I've got members of my own family who work in the public sector and local government so I know the pressures that they are under.
"I am confident, 100% confident that we will deliver a legal and balanced budget."
The local authority is now looking at where savings can be made and the public in the county are currently being asked for their thoughts on reducing the library service.
A spokeswoman for Denbighshire County Council said: “Denbighshire County Council, like local authorities across Wales, is facing a series of continuing budget pressures due to rising costs and demand for services.
“It is estimated that delivering day-to-day services – including social services, waste collection and schools – will cost an extra £26m due to price increases, inflation, and pressure on demand.
“Despite an expected increase in funding of £5.6m (3%) by Welsh Government, this still leaves a funding gap of £20.4m.
"The council must find additional money through savings and efficiencies, charges for services, increases in council tax, or by reducing or cutting services.”
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