Two in five council areas in England will not have enough primary school places for the number of children by September 2016, say council leaders.
The Local Government Association (LGA) says pressure from a rising population is reaching a “tipping point”.
Time was running out to avert a “national crisis” over a lack of places, it warned.
A Conservative spokesman said the government had created over 440,000 places and must “stick to the course”.
The rising number of pupils needing primary places has seen many schools having to expand and add extra classes, particularly in London and some of England’s major cities.
‘Run out of space’
The LGA now says that schools are “reaching their limits and could soon run out of space and money for extra places”.
It is warning that whoever forms the next government after the general election will have to tackle an escalating shortage of places.
“This tipping point is the biggest challenge the next education secretary will face and councils need a firm commitment that politicians will do everything necessary to ensure no child goes without a place,” said David Simmonds, the chairman of the LGA’s children and young people board.
Over the next decade, the local authorities’ organisation says there will need to be places for an extra 880,000 pupils at a cost of £12bn.
And they have highlighted the scale of the short-term problems.
In autumn 2016, the LGA says there will be a shortage of places for pupils wanting to start primary school in two in five authorities.
The following year half of local authorities will be facing a shortfall, says the LGA. By autumn 2018, this will have risen to three in five, according to the councils’ forecast.
To prevent pupils being left without places, the LGA says the next government will need to commit sufficient funding and give councils the flexibility to create new schools where they are most needed.
The LGA has been critical of the emphasis on opening free schools, when, the councils argue, they are better placed to make strategic decisions about where new places should be created.
“As we approach a new parliament, the next government must commit to fully fund the creation of all new school places and give councils the powers to open new schools once again, before time runs out and we are faced with a national crisis,” said Mr Simmonds.
“Councils face an uphill battle creating places where they are needed when their hands are tied by red tape and they are short of money to do so.”
Russell Hobby, leader of the National Association of Head Teachers, said there was a need for “unified local planning” to ensure that even with different types of state school there would be enough school places in each area.
Labour’s Tristram Hunt blamed the government for “ploughing hundreds of millions of taxpayers’ money on free schools in areas where there are no shortages of school places”.
He said the pressure on places was contributing to larger class sizes and children having to travel further to school.
The Tory party spokesman said: “The last Labour government failed to plan for the future, cutting funding for 200,000 school places during a baby boom. By contrast, this government has created over 440,000.
“As part of our long-term economic plan we are providing the best schools and skills for young people so that they can fulfil their potential and get on in life, and we must stick to the course.”
UKIP’s deputy leader and education spokesman, Paul Nuttall, said: “The more people we have living in this country, the more pressure there is on services.”