The controlling Labour group on Bradford MBC won a vote to allow more talks with the government and to see which companies are interested.
But Conservatives and Liberal Democrats both accused Labour of jumping the gun before the outcome of what is expected to be a critical Ofsted report.
They condemned the lack of consultation, saying the idea had come 'like a bolt out of the blue'.
The delivery of services, supporting schools to raise standards would be contracted out to the private sector. Labour says the change will make schools able to manage themselves more effectively and will mean more money going into education.
Council leader Ian Greenwood said: 'We do not believe the standards and results are sufficient for the district to grow and for the fulfilment of individual children.
'We do not blame individual schools. Nothing in these proposals will have any effect on governing bodies, headteachers, the employment of teachers. We are talking about our LEA services which supports schools and drives up standards across the whole district. We are convinced that has to change.'