Free schools for primary-age pupils have been set up in parts of the country where there is "no basic need", government documents say, despite figures warning of an unprecedented national shortage of 120,000 places in September.

Impact assessments of the first wave of free schools show that the Department for Education has privately recognised that primary schools have been established at a cost to the taxpayer in areas where there is no lack of places. In one case, a school was set up despite the acknowledgment that there was never likely to be a need for more primary places in the area.

The schools were opened despite a growing capacity crisis in other parts of the country, with headteachers being forced to convert music rooms to classrooms and build temporary facilities in playgrounds. Record numbers of children are expected to be taught in classes of 31 or more from this September.

A previous analysis of the catchment areas of the first 24 free schools, run outside of the control of local authorities, claimed that they are skewed towards middle-class children.

One impact assessment – on the Priors School, in east Warwickshire, a former private school that became a state-funded free school in September 2011 – warned of local concerns about "surplus places, pupil numbers and sustainability, consultation, use of taxpayers' money, standards, premises and parental choice".

"Demographic evidence shows that there is no basic need in the area," it said, "and data from Warwickshire and Northamptonshire county councils shows surplus places across the counties. The number of surplus places is forecast to reduce substantially in the next few years, although it is unlikely that there will ever be a shortage of places unless there are some school closures."

The report added: "Warwickshire county council is keen to emphasise that, while it is supportive of the free school policy in principle, it does not believe that there is a need for additional places in this area."

Another free school, Rainbow Primary in Bradford, opened despite the local authority telling the government that "these additional primary places are not required, especially in light of … plans to expand a number of existing primary schools and the expansion of some existing academies to all-through schools".

In Lathom, Lancashire, the Maharishi free school, where pupils receive what is described as a "consciousness-based education", expanded when it converted from the private sector to become a free school, despite the government's assessment that "the surplus of places among the five schools within a mile of Maharishi varies from 9%-23%. This equates to around 370 surplus places."

The series of impact assessments of the first wave of free schools was released by the Department for Education following a request under the Freedom of Information Act. The decision to back the opening of the schools was justified in each case by a desire to respond to parental demand and foster an improvement in standards.

A request for the impact statements was first made in October 2011 but the information was not published until 28 June. The information will raise questions about the government's priorities at a time when cash for school building is short.

The National Audit Office predicts that the primary school population will rise by 230,000 places from 3.97 million in 2011-12 to 4.2 million in 2013-14.

Even if local authorities achieve their plans of creating 110,000 additional primary places, as trumpeted by the education secretary, Michael Gove, Labour claims they will fall short of the 230,000 primary places that the audit office says are required.

A report from the Commons public accounts committee last week warned that Gove's "department does not sufficiently understand the risks to children's learning and development".

The report warns that "in areas where there is pressure on school places and a shortage of suitable land, authorities may have to convert communal spaces and specialist areas (such as libraries or music rooms) into classrooms".

Stephen Twigg, the shadow education secretary, said: "The real scandal is that Michael Gove is spending hundreds of millions of pounds setting up free schools in areas which don't have a shortage of places. This money could be used to sort out the school places crisis."

A spokesman for the Department for Education said: "The vast majority – 91 per cent – of the primary free schools approved last month are in areas of basic need. But the truth is that English schools have not been good enough for far too long. We make no apologies for encouraging new people to come forward to run free schools. The evidence proves that new schools also encourage the ones which already exist to raise their game.

"Free schools are proving highly popular with families who expect better than the old 'take it or leave it' offer they used to get from councils."