The general secretary of School Leaders Scotland claimed the number of pupils per class would also rise, with teacher vacancies to remain unfilled.
It comes as Cosla warned of job losses and service cuts in local authorities.
The government says it will work with local authorities to "protect what matters in Scottish society".
Deputy First Minister John Swinney laid out the draft budget on Thursday, announcing a change to Scotland's tax rates that will put more of the burden on those earning more than £43,000.
He also pledged a £1bn increase in health and social care spending as well as a £550m year-on-year increase for local government.
However, analysis published by Cosla suggests the cash increase could be just £71m once Scottish government-mandated policies are paid for, while it claims capital funding will remain flat for next year.
The organisation had called for a £1bn cash injection to maintain current services and fund pay deals for staff.
Jim Thewliss, general secretary of the School Leaders Scotland union, said the spending plans would lead to cuts in schools.
"We are now at a stage where we really have a service where if you're going to start to cut any more you're cutting into the basic functions of the service, which is essentially … the number of teachers in front of young people in schools," he told BBC Scotland's The Sunday Show.
"Vacancies which are already identified within schools - head teachers have been told these are not going to be filled."
Mr Thewliss said teachers across the country were expecting increases in class sizes and said some subjects would be removed if "deemed as non-viable".
He gave the example of higher modern languages subjects, which he said often had small class sizes of four or five pupils.
"We're now starting to find even greater pressure coming upon what is deemed as non-viable in terms of class sizes," the union chief told the programme.
He added: "We have bought entirely into the whole notion of young people having the best educational experience and the best life chances that they can have.
"We are now in the position … of managing expectations to a degree. And I've no idea where we're going to move with this but it's becoming much, much more challenging in managing those expectations at the same time as promoting aspiration."
Katie Hagman, an SNP councillor and Cosla resources spokesperson, said the budget announcement had been met with "extreme disappointment" after the organisation said it received less than a tenth of what it had asked for from the government.
She told The Sunday Show: "If we got the £1bn that would save, potentially, job cuts.
"What it translates to is equivalent of, say, 17,500 teachers. We're looking at essential services, things like youth work, things like economic development, working with our third sectors as well for domestic abuse and family support services.
"All these are considered essential services and we're at the front line of local government and it's these types of services that we will be desperately trying to protect."
Mr Swinney defended his budget announcement but acknowledged he had set out a "bleak picture" in parliament.
He told the programme that the war in Ukraine had increased inflation and the financial stress had been "compounded by some of the significant strategic mistakes" by the UK government.
He disputed claims the councils had been short-changed, saying their funding had increased by £570m year-on-year.
In response to warnings of service and jobs cuts, he said: "Local government came to me and asked for £1bn and I said up front there was just no way I was going to be able to achieve that. But I have managed to put £570m into the local government settlement.
"I've said to local authorities we need to work together to ensure that we protect what matters in Scottish society, which is preventative early intervention services where we do our best to change the outcomes for people within our society."
While the Deputy First Minister said councils would have to be "open to the concept of public sector reform", he insisted education was the government's priority.
"There's sustained investment being made in local authority expenditure, there is sustained investment being made in the Scottish attainment challenge, sustained investment in early learning and child care, key elements of how we deliver the curriculum and how we ensure people are well supported and young people are well supported to fulfil their potential," Mr Swinney added.
Jim Thewliss said the Scottish government's spending plans would lead to cuts in schools
Deputy First Minister John Swinney defended his government's budget