Unions have criticised Theresa May’s proposed public sector pay rises as “selective handouts” that are unlikely to benefit essential workers including teaching assistants, care workers and refuse collectors.

A £2bn public sector pay rise, the biggest in six years, is expected to be announced by the Treasury on Monday as one of May’s last acts as prime minister.

Only 2 million of 5.36 million public sector workers are expected to benefit and, with the exception of some extra funding for schools, the rises are expected to have to be drawn from existing budgets.

The GMB union said the alleged plans were “selective handouts” that will give nothing to support staff, most civil servants and people employed by local authorities.

The union’s national secretary, Rehana Azam, said: “Not only is this a pitiful pay rise, but it’s an insult to the millions of public sector workers it ignores.

“These are the people who look after us, help our children learn, and keep our towns and cities running.

The news was also met with concerns from headteachers over how the increases could affect schools, which are already struggling to cope with funding cuts.

Evidence submitted by the education secretary, Damian Hinds, to the School Teachers’ Review Body earlier this year, argued that schools could only afford a 2% pay rise for teachers – meaning that the remaining 0.75% would need additional funding.

Jules White, a secondary school headteacher in West Sussex who has led a national campaign to increase school funding, said his colleagues would be “dismayed” to learn most of the pay increase would come from current school budgets.

“Heads are being given Hobson’s choice. Do we fund a legitimate pay award – which may briefly stave off a crisis in teacher supply – or do we try to use our meagre budgets elsewhere?” he said.

“We need real-terms investment across our whole system rather than being expected to lead schools on a hand to mouth basis.”

Under the plans, police officers will receive a 2.5% rise and the salaries of teachers and other school staff will increase by 2.75%.

Soldiers will also get a 2.9% increase while dentists and consultants will get 2.5%, according to the Times.

Most of the pay increases given to public sector workers will be higher than the 2% rate of inflation. Many have been forced to accept below-inflation pay increases over the last decade as part of the government’s austerity cuts.

Gail Cartmail, assistant general secretary at one of Britain’s biggest trade unions, Unite, argued that public sector workers were being used as “a political football”. She argued that while all should benefit from the rise, the current proposal would not be enough to catch up with pay after years of pay freezes.

“The idea that police officers would think that this was the best thing since sliced bread given the huge pressure on police … It just doesn’t add up,” said Cartmail.

• This article was amended on 25 July 2019 because Unite is one of Britain’s biggest trade unions, not the biggest as an earlier version said.