Scape Group's research study, 'School Places Challenge', states more than 2,000 new schools must be built along with 24,000 extra classrooms over the next four years.
Local authorities are expecting an additional 729,000 pupils in education by 2020, a rise of 8.6% in primary school pupils and 12% in secondary school pupils in England alone.
London, the South East and East of England are experiencing the highest growth with more than 375,000 additional primary and secondary pupils expected to be added to the registers.
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Rapid growth in the cities of Bristol, Peterborough, Milton Keynes, Leicester and Nottingham will also mean that new schools will be in high demand should the necessary amount not be built in time.
Mark Robinson, Scape Group Chief Executive, said: "As the growth of the primary school population gathers pace, the pressure on school places will soon transfer to the secondary population, requiring a new wave of advanced school building.
"The government's preference for free schools has created uncertainty for local authorities, who are tasked with planning and building new schools, but will not be responsible for running them. Proposals for new grammar schools has further muddied the waters.
"Leaving the EU could have a profound effect on the UK population – we could see a short term surge in migration from Europe or a drop off in numbers if migrants are put off coming here. It is very difficult to predict what impact Brexit will ultimately have, putting even more pressure on local authorities who are planning how and where to prioritise school-building."
(LM)
