The findings come as children's secretary Ed Balls prepares to announce an extra £200m to fund more primary school places.
The LGA said a reduction in the number of parents paying for private education could have caused the pressure on places.
It also claims that families who may have intended to move out of major cities when their children reached school age are now unable to sell their homes due to a standstill in the housing market.
The number of school places needed within a local area is predicted from year to year using population forecasts and census information. But according to the LGA unpredictable events, such as the recession, can affect the reliability of predictions.
Les Lawrence, chairman of the LGA's Children and Young People's Board, said: "Councils that are being affected by shortages for next September will be aware of the problem and will be reviewing their data for the coming years.
"In the short-term councils will be working with schools to help them find extra capacity and draft in extra teachers and support staff, but there also needs to be an emphasis on improving methods of forecasting for the future."
Last month, CYP Now reported that councils in London were struggling to find the cash to meet a surge in demand for primary school places.
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