The party claims that 40 councils in England will have more pupils than places for the next school year.
Their analysis is based on School Capacity figures for May 2013.
In March, the National Audit Office estimated that 240,000 new primary school places would be needed by September 2014 to meet increased demand.
And the problem is set to get worse, says Labour, with nearly half of councils forecasting a shortage of places for September 2015, rising to two-thirds of councils by 2017.
A Freedom of Information (FOI) request submitted by the party has also revealed that a quarter of councils believe their current level of Government funding is not sufficient to create the extra school places needed to meet demand for 2015/16.
One quarter of councils that replied to the FOI request also reported that they did not think the Department for Education’s cost assumptions for primary school places in the local authority were realistic.
Shadow education secretary Tristram Hunt said, ‘David Cameron is failing in his most important duty to children in this country: to ensure every child has a place at a good local school. He has created a crisis in school places, ignoring the warnings and diverting millions of pounds away from children in areas where there are shortages of places.
‘This is damaging standards in our schools, with class sizes soaring, more temporary and unsuitable classrooms and children forced to travel further and further to get to school.
‘A future Labour Government will have the right priorities for driving up school standards. We will prioritise new school places in areas where there are shortages, have rigorous local oversight of schools and ensure that all teachers have or are working towards qualified teacher status.’
However, the Department for Education (DfE) claimed that Labour’s figures were ‘bogus’.
A DfE spokesperson said, ‘These figures are based on a completely false comparison and would only be true if local authorities had taken no action to create places since 2012-13, despite the huge amounts of money we have given them.
’We are giving councils £5 billion to spend on new school places over this parliament — double the amount allocated by the previous Government over an equivalent period.
’This funding has already led to the creation of 260,000 new school places, all of which are in areas where there is a shortage of places. All councils should now have plans in place to meet the need for this September and we will hold to account any that fails to do so.
‘In addition we have also confirmed a further £2 billion for basic need up to 2017. Investment in free schools is on top of this funding, and in fact seven in ten free school places are in areas of basic need.’