Labour has accused the government of creating a "crisis in school places" after publishing an analysis suggesting that the demand for primary school places at the start of the academic year in September will be greater than the supply in more than a quarter of council areas in England.

Freedom of information responses from 131 local authorities showed that 32 – a quarter of the total – believe that government funding for additional primary places is insufficient to meet demand for 2015/16, and 33 thought Department for Education (DfE) cost assumptions for new places were not realistic.

But the DfE dismissed Labour's figures as bogus, arguing that they did not take into account new places created since 2012/13 – the latest year for which figures are available. A spokesman said the government was providing £5bn for new primary places over the course of the parliament, which had already resulted in the creation of space for 260,000 more children.

The row comes days before parents find out whether their children have got places at their chosen schools on April 16.

Figures released by the DfE shortly before Christmas showed that 3,514 schools across England had more pupils than their official capacity last May, but no councils reported having more pupils than places overall.

However, with primary school rolls expected to rise sharply in the next few years, 40 councils (26% of the total) were forecasting that they would have more pupils than their 2012/13 space provision by the start of the 2014/15 academic year, rising to 70 authorities (46%) in 2015/16, 93 (61%) in 2016/17 and 101 (66%) in 2017/18.

Labour argues that money has been diverted away from areas threatened with shortages by the government's focus on the free school programme, following a National Audit Office report which found that £241m of the £950m projected capital spend on mainstream free schools was in "districts with no forecast need for extra school places".

The 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 a DfE spokesman said: "These figures are bogus. They 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 £5bn 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 confirmed a further £2bn for basic need up to 2017. Investment in free schools is on top of this funding, and in fact seven in 10 free school places are in areas of basic need."

Labour said that the pressure on school places had resulted in the number of "titan" primaries with over 800 pupils more than doubling from 16 in 2010 to 43 last year, while the number of infants taught in classes of over 30 over the same period increased from 31,265 to 71,935.