The Local Government Association says councils are now providing temporary housing for 120,540 children with their families.
It says the growth rate - equivalent to an extra secondary school's worth of children per month - is unsustainable.
The government said the figures were a worry but still below the peak of 2006.
Based on the latest figures from the Department for Communities and Local Government from March 2017, the figures show a net increase of 32,650 (37%) since the second quarter of 2014 - an average of 906 extra children every month - just 40 fewer than the number of pupils in the average secondary school.
Councils say the net cost of providing temporary accommodation has tripled in the last three years, as the extra demand for places increases pressure on local government.
And the LGA, which represents 350 councils in England, says councils need to be able to build more genuinely affordable homes and provide the support that reduces the risk of homelessness in the first place.
This means councils being able to borrow to build and to keep 100% of the receipts of any home they sell to reinvest in new and existing housing.
Council leaders are also calling for an adaption to the implementation of welfare reforms to reduce the risk of homelessness and for access to funding to provide settled accommodation for families that become homeless.
Councillor Martin Tett, the association's housing spokesman, said: "Whilst the government's indication [that] it will explore ways to enable councils to build more homes is encouraging, these new homes can't appear overnight, and the demand is urgent.
"Councils are working hard to tackle homelessness, with some truly innovative work around the country - and we now need the government to support this local effort by allowing councils to invest in building genuinely affordable homes and taking steps to adapt welfare reforms to ensure housing remains affordable for low-income families."
The LGA sets out the lengths that councils are going to in order to tackle homelessness in their area in a new report.
Examples include innovative modular housing, dynamic purchasing systems and offers of housing in private rented sector.
A DCMG spokesman said: "We're clear that whilst temporary accommodation is vital in making sure that no family is without a roof over their head, councils have a responsibility to find secure good quality accommodation as quickly as possible.
"This government is determined to tackle homelessness and rough sleeping - that's why we're investing Â£550m to help tackle the issue."
The number of homeless children being housed in temporary accommodation rose by more than a third in the last three years, according to official figures.