Fourteen are community and voluntary nurseries, but two private nurseries, three on school sites and one NHS nursery are also affected.
The council claims that a cut of £6.8m to the Early Intervention Grant forced it to take the decision to end funding at the end of March.
Parents and supporters have waged a long campaign opposing the cuts with more than 10,000 people signing a petition and holding a number of protests.
The council insists that it will support settings with business plans so that they can stay open.
No nurseries have closed yet but campaigners fear that some will close or be taken over by private providers.
Families from the affected nurseries have sought legal advice and are in the process of applying for a judicial review, with the aim of overturning the local authority's decision and having the grants reinstated.
Douglas Johnson from the Sheffield Law Centre, who is acting on behalf of parents of children attending some of the nurseries, said, 'We're hoping to issue a judicial review shortly. We've given the council lots of chances to respond and put action plans in place for each of the nurseries affected. They've failed to do that. We're now preparing to issue proceedings as part of a judicial review.'
The council heard evidence at a scrutiny board meeting at the end of February, but councillors voted to go ahead with the cuts.
Early Education submitted a discussion paper to the board. Di Chilvers, chair of Early Education in Sheffield and a trustee of Tinsley Green Community Nursery and Children's Centre, told Nursery World, 'We have the biggest membership outside London. We wanted to be part of the discussions and provide an objective view.'
But Ms Chilvers said she felt the scrutiny board did not listen to the evidence. 'It was an inhospitable meeting, chaired aggressively and with little time for people to make their case and no real discussion.' She said she was dismayed at the speed at which the council had cut the grant and believed that although some nurseries would struggle on and survive, others would be forced to close.
Redundancy notices have already been issued to staff at Tinsley Green and at some other settings. 'They are doing their very best to look for other ways to keep going and two-year-old funding might help, but they will still need some funding.' Recent DfE research concluded that public funding is needed to provide childcare in disadvantaged areas and that private provision alone is not sustainable.
'Provision is being dismantled so rapidly in Sheffield as a response to the cuts,' said Ms Chilvers. 'It would be naive to think there would be no cuts, but councils have powers of their own to decide how to spend the funding they have and they could look at ways to work more smartly with the funding they have. Other neighbouring LAs are doing their utmost to keep their nurseries and children's centres open. I believe that there are other models they could have looked at with comparable areas.'
The council is offering places for two-year-olds from this month. But Ms Chilvers believed that it would struggle to meet its sufficiency duties for providing two-year-old places to the most disadvantaged children, because of the 'very real geographical divide in Sheffield between wealthy families and families living in poverty.'
While Nick Clegg's constituency in Hallam is one of the most well-off there are others with very high levels of poverty, such as Manor and Darnall.
Many parents using free early education places in the poorest areas only use the free entitlement and do not pay for more hours.
Ms Chilvers said, 'The council has said that places will be provided by private providers, but why would a private provider move into some of these areas? They will think twice because of the impact on their business.'
She said that it was difficult to travel from Tinsley Green by public transport to other parts of the city. 'If you're a young mum with a two-year-old, you're not going to take two buses to get to a nursery.'
Ms Chilvers is also concerned that the council has said that it will support schools to take two-year-olds. 'That is very worrying because a lot depends on the school and the quality of their provision and practice. Research has shown that only high-quality settings will support vulnerable two-year-olds and direction from the Government is that only good or outstanding settings can take these children. In many cases schools are struggling with three-year-olds, let alone two-year-olds.'
Ms Chilvers also sees a threat to good early years practice. Five of the community nurseries that have lost their funding are outstanding settings. 'It's taken a long time to build up good practice. My concern is that there will be a pedagogical drain and we will lose all that.'
The council said it was committed to making sure there would be enough childcare places for families that need them.
The council is also cutting the number of Ofsted registered children's centres to 17. Services will continue to be delivered 'in the same number of outlets' across the city, but may be in different buildings.
Councillor Jackie Drayton, cabinet member for children, young people and families, said, 'We are not closing children's centres. Services will continue in the same number of areas they currently do. However, they may be delivered in different venues, at different times and by different providers.'