The LGA said that it had found only “a handful of cases” of local authorities involved with Bernard Madoff's schemes.
Hampshire County Council is one and expects losses of around£7.1m of the authority's pension fund - 0.3% of the fund's total assets of£2,400m.
Ken Thornber, the leader of Hampshire County Council, said there was no risk to the Fund's 46,000 contributors or 27,000 pensioners, both in terms of existing pensions and future benefits.
He said: "The actual position remains to be seen and Hampshire's exposure is extremely small relative to other investors. Nevertheless the Hampshire Pension Fund is very concerned that a small proportion of its investments may be at risk because of this alleged fraud."
Wall Street veteran Mr Madoff’s was arrested after reportedly confessing to employees that the business that bears his name was a giant pyramid scheme.
A LGA statement said: “The only exposure identified so far has been through money invested by local authority pension funds. Potential losses currently identified are small, and relate to only a small percentage of overall investments. Pension fund investments are held separately from the rest of the council's money.”
LGA chair Margaret Eaton (Con), said: “We are still investigating the extent of councils’ exposure to these investments, which seems likely to be limited to some councils' pension funds. Councils will be confident that this won’t have an impact either on council tax or on local services.
“Prudent financial management means that councils put their money into a diverse range of investments to make sure that any risk is spread.
"We are not aware of any councils that are in serious problems and we are confident that vital frontline services will remain unaffected.”