Many parents are set to be "disappointed" today when their children are rejected for the school of their choice, a Government minister has warned.
Priti Patel said it was "deeply regrettable" that families in England would be hit by a shortage in primary school places.
The employment minister, a Brexit supporter, blamed uncontrolled migration for putting "unsustainable pressure" on public services and warned the problem would only get worse as more countries became members of the European Union.
Hundreds of thousands of four and five-year-olds will learn which primary school they will be attending from September, on what is commonly known as National Offer Day.
Ms Patel said: "The shortage of primary school places is yet another example of how uncontrolled migration is putting unsustainable pressures on our public services. Education is one of the most important things that Government delivers, and it's deeply regrettable that so many families with young children are set to be disappointed today.
"The truth is that for as long as we remain a member of the EU we are completely unable to control the numbers of people coming to this country - and with another five countries in the pipeline to join the EU the problem is set to get even worse.
"If we Vote Leave we can take back control of our borders."
Unions and town hall leaders warned Government reforms that mean all schools will convert to academies are set to fuel the shortage in school places.
Councils will not have the power to force schools to expand in the future, even where there is demand and capacity, the Local Government Association (LGA) said.
It warned that an additional 336,000 primary school places would be needed by 2024.
Roy Perry, chairman of the LGA's children and young people board, said: "Most academies will be keen to work with their local authorities, but in the minority of situations where this isn't the case, appropriate powers are vital to ensure all children get a suitable place."
The Association of Teachers and Lecturers (ATL) said it was "worried" about the impact of the reforms.
Nansi Ellis, assistant general secretary, said: "Many parents already find it stressful to find a school place for their child. And we fear parents will find it even harder to get their child into a school of their choice, especially if they are trying to get all their children into the same school, if all schools become academies.
"We are already hearing that 90 primary schools have a catchment of only 330 yards from the school gate, and the number is likely to rise as the school population increases."
Lucy Powell, shadow education secretary, said: "This Government's broken school places system means that children are being crammed into ever larger class sizes and many schools are already at or over capacity."
She added: "Ministers have already tied the hands of local areas to adequately plan for school places. The Tories' new forced academisation policy will make the school places system implode, as councils lose completely the levers they have to ensure there are enough school places for children."
The Government spent £5 billion creating places between 2011 and 2015 and 95.9% of parents received an offer at one of their top three preferred primary schools last year, the Department for Education said.
"Despite rising pupil numbers, at primary, the number of pupils in excess of their school's capacity has fallen by a quarter since 2010, and average class sizes have seen little change," a spokesman said.
"Of course there is more to do - that's why this Government has already committed to invest a further £7 billion to support councils in delivering school places, which along with our investment in 500 new free schools we expect to deliver 600,000 new places by 2021.
"It is simply not true to suggest councils cannot commission new schools – where councils identify that a new school is needed in their area they are required to run a competition to identify strong providers for a new free school."
A Whitehall source said: "There's no evidence that migration is the key driver of demand for primary school places."