Almost half of school districts in England will have nearly twice as many pupils as available primary school places in two years, council leaders have warned.
As many as two in three councils could see more children looking to start primary school in their area by September 2016 than there are currently places for, the Local Government Association (LGA) said.
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Schools are already having to convert non-classroom space, such as music rooms and libraries, into classrooms and others have been forced to reduce playground space or expand class sizes, the association warned.
The warning comes as the Government announces 93 new free schools opening across the country, creating an extra 43,000 spaces for primary and secondary school children.
The new schools will bring the overall number of open free schools to 174.
Of the 93 schools opening this month 35 are primaries, 42 are secondaries, 11 are all-age schools and five will cater for 16 to 19-year-olds.
The LGA is calling for the Department for Education to work more closely with local councils, so planning for emerging demand for places can be better managed.
Its analysis of local authority data suggests about 1,000 of the 2,277 local school planning districts will be over capacity by 2015-16, the BBC said.
Education Secretary Michael Gove says there is "definitely an issue" ensuring enough primary school places across the country, but that the government inherited a "terrible" situation from Labour.
"We have significantly increased the funding for primary schools. No-one is going without a school place this September," he said.
However, Labour's Shadow Education Secretary Stephen Twigg has blamed Michael Gove's decision to end primary school building for the expected shortage of primary school places.
"I've seen Michael Gove grand-standing today, suggesting somehow Labour can be blamed," he said.
"The root of this problem was the decision Michael Gove and David Cameron took in 2010, to end Labour's programme of primary school building," he added.