It said in a report that cuts had not been applied equally since 2010, with local authorities in the poorest areas seeing the biggest reductions.
The MPs warned further cuts could undermine councils' financial stability and threaten statutory services.
Local authority funding has been cut as part of efforts to tackle the deficit.
Most local authority funding in England comes from central government, with about a quarter of it raised through council tax.
Public spending watchdog the National Audit Office has said that by 2016 local authority funding will have been cut in real terms by 37% since 2010.
In a report on the financial sustainability of local authorities, the Public Accounts Committee said that while local authorities had responded well to cuts "on the whole", there were concerns over whether some councils would continue to be financially sustainable.
"This is particularly the case for authorities responsible for adult social care and children's services," it said.
Committee chairwoman Margaret Hodge, a Labour MP, said: "These cuts have not hit all local authorities equally, with reductions ranging between 5% and 40%.
"Councils with the greatest spending needs - the most deprived authorities - have been receiving the largest reductions."
The report said the Department for Communities and Local Government - which has overall responsibility for council funding - did not fully understand the impact of funding cuts.
It said the department relied too much on spending data and had insufficient information on service levels, service quality and the financial sustainability of councils.
"Without at least an idea of the amount of funding required to maintain statutory services to a minimum standard, it is hard to see how the department could ensure that local authorities are able to fulfil their statutory duties," Mrs Hodge said.
The report also questioned whether the department was providing effective enough leadership to ensure councils could change the way they deliver their services to absorb future cuts and remain financially sustainable.
"Overall, as pressure from cuts grows, so do the risks to local authorities' finances and their provision of services.
"The depth and quality of the department's insight into these issues needs to keep pace with these changes, something it has struggled so far to achieve," the report said.
In December, ministers outlined an average funding cut of 1.8% in English councils' overall spending power, as part of the local government finance settlement for 2015-16.
Some councils will face cuts of up to 6.4% - which local authority bosses said would "push some authorities to breaking point".
The government says the funding grants settlement was "fair for all parts of the country".
Local Government Minister Kris Hopkins said at the time that the settlement still left councillors with "considerable total spending power", and that many councils were seeing growth in income from business rates.
Councils in the most deprived areas of England have been hardest hit by cuts to their funding, the Public Accounts Committee has said.