Humza Yousaf has been warned town hall bosses will only support a council tax freeze if they are allowed to cut teacher numbers to balance the books.
Council chiefs also said national plans to protect the time teachers spend in front of pupils need to be axed.
Andrea Bradley, general secretary of the EIS trade union, said: “It is simply nonsensical to suggest that reducing the number of teachers in our schools, or cutting the length of the pupil week, is in any way linked to improving educational outcomes.”
They fear a freeze will be funded on the cheap and result in councils having to make deep cuts to lifeline services.
A letter to Yousaf from Shona Morrison, who is President of council body COSLA, repeated the frustration felt over the freeze and set out a list of conditions for backing the policy.
One of the biggest asks is councils making their own decisions on teacher numbers.
Local authorities wanted to cut posts earlier this year to protect other services, but this was prevented by former First Minister Nicola Sturgeon.
In her letter, Morrison called for the “removal of arbitrary teacher numbers”.
She wrote: “Improving outcomes for children and young people cannot be provided by teachers alone.
“It requires a range of supports and services that should be for local determination (including social work support, early intervention programmes for children and families, early years, school transport, breakfast and after school clubs).
“The funding that Scottish Government has provided in the settlement should be able to be used to provide the very best support for our children and young people, whether that be a teacher, a pupil support staff, or family support.”
Yousaf’s Government is also consulting on plans to set a legal minimum number of hours on the education pupils receive each year.
This was prompted by concerns that councils were proposing to cut class contact time to plug massive funding black holes.
Morrison told Yousaf “restrictions” on the make up of the school week should be binned if the council tax is frozen.
She wrote: “The Scottish Government's proposals for minimum learning hours do not reflect the need for flexibility to meet the diverse needs of Scotland's communities or allow councils to draw together services to maximise impact whilst reflecting and shaping local places and the economy.
“Until robust evidence is confirmed on the link between learning hours and outcomes, councils should be able to configure the working week to suit local communities and families.”
The Record revealed in January how Glasgow Council wanted to slash 800 teaching jobs and close primary schools early on Fridays to help save £51m.
Labour MSP Pam Duncan-Glancy said: "Instead of being able to take time and make the best decisions for the future of education, councils are being forced to react to the SNPs chaotic policy making and rushed choices.
"The fact that councils even have to consider decisions like these shows how little support they are getting from the Scottish Government and proves again that the SNP can’t be trusted on education.
"Education should always be a top priority, but SNP incompetence has left all of that in chaos.
“Only a Labour government can stand up for our schools and ensure every child gets the opportunities they deserve."
Scottish Lib Dem MSP Willie Rennie said: “Education has become the SNP’s defining failure. Having stripped schools of well-qualified teachers throughout their 15 years in power, the nationalists’ shambolic stance on council tax threatens to make those problems even worse.
“If this freeze is to be imposed, councils will need to be compensated in full to prevent an axing of vital services. That should be backed by a proper deal for local government that will reverse years of budget cuts and introduce a fair alternative to the council tax.”
As well as tying support for a fully-funded freeze to flexibility on education budgets, Morrison asked for reassurances on the council tax policy.
She wanted confirmation it is for “single year” only and that councils will not be penalised for refusing to freeze the tax.
Another COSLA ask is for realistic timescales on the controversial plan for a national care service.
Morrison also demanded clear progress on council tax reform next year.
She added: “Councils are key to achieving better outcomes for the people of Scotland.
“We believe this has been overlooked in recent years and that Scottish Government has failed to recognise the early intervention and prevention role that only councils can play.”
A Scottish Government spokesperson said: “The Council Tax freeze next year will benefit every Council Tax-payer in Scotland at a time when rising prices are putting significant strain on household finances.
“The total funding will be discussed and agreed in partnership with Local Government, and will form part of broader funding decisions made in the 2024-25 Scottish Budget.
“Teacher numbers are at historically high levels and Scotland has the lowest pupil teacher ratio in the UK.
"We have provided councils with an additional £145.5 million in this year's budget to protect increased teacher numbers, recognising the vital importance of teachers in our classrooms and in line with our clear commitments to deliver excellence in Scottish education."
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