Starting this week Peterborough CEO Gillian Beasley is dividing her time between Peterborough and Cambridge. Ms Beasley has already stated that for Cambridgeshire County Council job cuts are underway.
The Conservative peer said: "We know we've got probably 12 or 14 councils that are very close to the edge now."
The government said councils had £22bn in their reserves and had to play their part along with the rest of the public sector in tackling the deficit.
Councils are not allowed to run deficits. The LGA declined to reveal which councils were affected.
Chancellor George Osborne has asked government departments to draw up plans for cuts of 40% and 25%.
The LGA released to the BBC an analysis which said 40% cuts would leave councils £20bn worse off.
Lord Porter told BBC Radio 4's PM programme some councils "will be very close in the life of this Parliament to not being able to meet their statutory obligation to balance their budget".
He said: "We're not shroud-waving - we kind of know we need to do more and better and do it with less money - but we're saying there are better ways of doing things with less money than simply top-slicing some things out of local government."
A 40% reduction to core central government funding for councils in England and Wales would be worth £8.4bn while the same cut to separate local government grants would see a further £2.1bn lost, the LGA said.
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