Schools could be pushed to breaking over the next decade by the need to provide almost a million more places for pupils, town hall chiefs have warned.
The continuing squeeze on places could lead to a "tipping point" where there is no money or space left to expand schools any further, according to the Local Government Association.
Official figures predict that there may be around 900,000 extra pupils in England's schools over the next decade.
The LGA said its own analysis had concluded that it will cost £12 billion to create enough school places for all of these children. It urged the government to ensure that all school places are fully funded and to enable councils to open new schools according to the needs of the local community.
The call comes just days before the January 15 deadline for parents to apply for primary school places for this September.
There have been ongoing concerns about pressure on school places, particularly for primary-age children, fuelled in part in recent years by a rising birth rate.
The government has pledged £7.35 billion so far to create more places, but the LGA argued that this still leaves a major shortfall.
Around 90,000 places were created in 2012/13 by councils, and more are being established - but the scale of the problem is too big to be solved at a local level, the Association insisted.
David Simmonds, chair of the LGA's children and young people board, said councils and schools were, in some cases, going to "extraordinary lengths to create places".
"But we fear a tipping point could soon emerge when councils and schools can no longer afford the massive costs for the creation of places, nor find the space necessary for new classes, if this crisis is not properly dealt with."
He added: "We do not want any child to be without a place. That is why, at a time when parents are making big decisions about their child's future, we are calling on the government to commit to funding the creation of school places and hand councils the powers to open new schools, for both primary and secondary-age pupils, before time runs out."
In 2013, a study by the National Audit Office (NAO) warned that by September 2014 an estimated extra 256,000 primary and secondary school places would be needed to meet demand. Of these, 240,000 were required in primary schools, with more than a third (37%) needed in London alone.
Last year, research by TES concluded that hundreds of thousands of pupils could end up with less room to play outside because a squeeze on school places has meant classrooms are being built on playgrounds and playing fields.
Conservative minister Nick Gibb laid the blame for the shortfall at the previous Labour administration's door.
"The last Labour government failed to plan for the future, cutting funding for school places during a baby boom while allowing immigration to get out of control, and they wasted millions of pounds of taxpayers' money on their failed Building Schools for the Future programme," he said.