Last month at the High Court in Manchester, Judge Bird granted permission for the parents to proceed to a full hearing of the judicial review.
In his ruling the judge said that if he refused permission, 'There could be a real risk of causing real injustice.'
As we reported in April, the council has stopped £3.6m in grants to the nurseries, putting their survival at risk.
Sheffield City Council has claimed that a cut of £6.8m to the Early Intervention Grant forced it to stop the funding at the end of March.
The two-day court hearing takes place on 29 and 30 July and a judgment is expected soon afterwards.
Douglas Johnson from the Sheffield Law Centre, who is acting on behalf of four women who all have children at the nurseries, told Nursery World, 'What we're asking the court to do is to quash the council's decision to cut the grants.'
The judge granted the judicial review based on five grounds. These are: inadequate consultation; a breach of the public sector equality duty, breach of duties under the Childcare Act, breach of the Best value duty - guidance says that the council should not cut funding to the voluntary sector more than to its own services, and that the decision was irrational.
One parent, who has four children, said, 'I have three children in the centre, one in the nursery and two in the Out of School, and I have just given birth to a baby who I had planned to have in the nursery as well. Places have been shut down due to the cut and now I'm in trouble as I was to return to work early.
'The council said there are places in other settings but there are not, I rang around. The consultation for these cuts was disgraceful and families complained in bucketloads, but nothing happened. Everyone was ignored. We even put in a 10,000-signature petition, opposing what the council was doing and wanted to do but this was not given any proper consideration and not taken on board. It was the biggest petition the council had ever been given, we were told.'
Sheffield's early years champion, Professor Pat Broadhead, (pictured), has resigned from her post over the issue. Professor Broadhead was the first person to hold the post.
Ms Broadhead, Professor Emeritus of Playful Learning at Leeds Metropolitan University told Nursery World, 'I felt really pleased when I was asked to do this and in the beginning I felt I could act as a bridge between service users, providers and local authority policy development. But at the time I resigned I felt this was no longer possible. I feel that Sheffield needs an early years champion who could feel more comfortable with the way things are moving.'
Professor Broadhead added, 'My biggest concern is that decisions that are now being made are having the greatest detrimental impact on the poorest parts of the city.'
She said that the local authority was making it very difficult for early education and family support services to continue when these are integral to their communities.