Stoke-on-Trent City Council said raising council tax by 3% would pay for extra social care and launched a consultation on its budget proposals.
The council, led by a coalition of City Independents and Conservatives, also said 202 jobs could go to save £61m.
Labour councillors criticised the budget as "attacking the most vulnerable".
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The authority last increased council tax rates in 2012.
The 3% rise means council tax for a Band D property would increase from £1,183.46 in 2016-17 to £1,218.96 in 2017-18.
The plans could see six children's centres moved into other council buildings or schools, which the council says would save £3m.
Other proposals include a £1.5m refurbishment of Hanley market and building a £9.5m multi-storey car park near Stoke railway station and Staffordshire University.
It also says raising rates will help to pay for 230 carers to help look after older people, while more than 100 of the planned redundancies will be from the Co-Operative Working scheme set up in February to help vulnerable people.
Council leader David Conway said the council was not closing children's centres and that said raising tax rates was "the right thing to do".
"We have to save £61m over the next three years - we have to do it, we have to manage the business," he said.
Mohammed Pervez, leader of the Labour opposition and former leader of the authority, said the budget would reward more affluent parts of the city instead of helping its poorest residents.
More than 200 jobs could be lost and council tax could rise under budget plans.