Council chief Ian Floyd said the organisation has an £8 million hole in the budget.
He said the council is consulting with trade unions and staff affected by plans to cut jobs, which could lead to "20 or more redundancies over a 90 day period".
The council was asked what departments could be affected by the redundancies and if frontline workers' jobs are at risk, but declined to say.
Mr Floyd, chief operating officer at City of York Council, said: "We are currently collectively consulting with our trade union colleagues and those staff affected on a number of proposals and as such have made the appropriate notifications as these proposals could result in 20 or more redundancies over a 90 day period.
"Given the potential implications of these proposals, it would not be appropriate to cause undue concern by speculating.
"However, as always, we will be strictly following the required policy and procedures for restructuring within the organisation, and this will ensure that we adhere to all necessary employment legislation."
Government figures published last week show York council is among a minority that are not facing a shortfall in council tax income amid the coronavirus pandemic.
Ministry of Housing, Communities and Local Government figures show that the council is one of 51 councils that has avoided the ‘financial hole’ caused by a drastic national fall in anticipated council tax income for the previous financial year.
York council budget papers published in February proposed more than £1.4 million in savings linked to staffing reductions, reviews and efficiencies in the next two years.
A budget meeting heard the council had already put strict recruitment controls in place, including choosing not to fill some vacancies, in order to minimise the impact.
The council is looking to make £8 million of savings as a result of the extra costs and drop in income, for example from car parking fees, caused by the pandemic.
Mr Floyd added: "Over the last year demand for council services has increased, and at the same time, income has significantly fallen as a result of the pandemic.
"Whilst the government has provided some financial support, it doesn’t not go far enough to cover the gap caused by falling income and increasing costs, meaning the council faces a significant budget gap.
"Recognising the current economic challenges and support needed for local residents, the council is increasing its total spending by £2.6 million on adult social care and support for children and young people, whilst investing additional resources to protect frontline services and to accelerate the city’s economic recovery."
"To help cover this funding gap, we are working with managers to see how we can achieve £8m of savings."
Government must be notified when an organisation is planning to dismiss 20 or more employees at a single establishment.