More than two-thirds of London councils are struggling to provide sufficient primary school places to meet demand, a BBC London investigation has found.
Classes are full in at least 24 of the 33 local authorities and the councils claim they face a collective shortfall of £500 million in government funding.
They say 7,000 extra places are needed across London over the next four years.
The government said there was record investment in education and schools and councils must "use the money well".
But the situation was "very serious", according to Jason Stacey from London Councils, which lobbies the government on behalf of the 33 local authorities.
"There is simply not the money in our capital budget to pay for expanding our primary schools," he told BBC London.
"Therefore unless we have government assistance, we are going to have two things - councils having to borrow lots of money in order to fund this, or we're going to have parents not able to get their children into primary school."
'Very, very concerned'
Only two boroughs have secured additional government funding to pay for the temporary classrooms needed to house the extra pupils.
Others have been spending money which was supposed to go towards new buildings or the refurbishment of existing schools.
Councils needed to "use money well", Schools Secretary Ed Balls insisted
The rise in demand for places is being attributed to the increasing birth rate in London.
There have been 21,000 more births in the past six years than had been predicted, figures from the Office of National Statistics suggest.
And the National Union of Teachers fears that having bigger class sizes will have a detrimental effect on children who do manage to find places in schools.
"We're very, very concerned about it," said Tim Harrison, the union's London secretary.
"In the past, this has meant things like temporary classrooms and it's meant oversized classes."
But Schools Secretary Ed Balls said more education funding than ever was being allocated to London's councils.
"My challenge to local authorities is use this money well so that you can raise standards in every school for every child," he told BBC London.
"There's record investment, but to justify keeping that record investment coming in, as I am determined to do, you've got to show results.
"That's down to the schools and local authorities in London to really use the money well."
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