English councils have called on the government to provide a £5bn "income guarantee" to prevent local authorities from having to cut services as the coronavirus hits finances.
The County Councils Network has warned councils could lose £2.4bn as income from tax and business rates fall.
The group said councils would have "no choice" but to suspend non-essential expenditure and cut services.
The government says it has provided "an unprecedented £3.2bn" to councils.
A spokesman said it was giving councils "the resources that they need to tackle the immediate pressures they have told us they're facing".
"This is on top of English councils' core spending power rising by over £2.9 billion this financial year and a further £600 million to help reduce the infection rate in care homes," he added.
The County Councils Network (CCN), which represents 36 county and unitary local authorities in England, says its members could lose an estimated £430m from a fall in fees and charges.
It also says councils stand to lose up to £2.4bn if the proportion of people unable to pay council tax rises to 20% and income from business rates falls by 10%.
Last month the Local Government Association said the income base was "collapsing" for councils, with leisure centres shut, public transport cut and parking fees not coming in.
The CCN warns that councils may have to suspend non-essential expenditure and implement cutbacks to services "including those aimed at fighting the spread of coronavirus".
To prevent this happening, the CCN says the government should make £5bn available to compensate councils for their lost income, pointing out the government wrote off the NHS' £13bn debt following the outbreak of the virus.
Cllr David Williams, chairman of the County Councils Network, said funding provided by the government was "very welcome and provides vital resources to meet immediate cost pressures, recognising that councils have done much of the heavy lifting during this pandemic, from protecting the vulnerable and taking the strain off the NHS".
He added: "However welcome one-off injections of resources have been, councils cannot budget on verbal reassurances alone and therefore now is the time for the government to step forward with firmer financial guarantees to stem fears that councils will have to declare insolvency."
The Local Government Association reiterated the message, saying that without "ongoing and consistent funding... councils and the services our communities rely on will face an existential crisis".
Cllr Richard Watts, who chairs the Local Government Association's resources board, said: "Some councils have warned that they will soon face the prospect of Section 114 reports - this would lead to spending blocks and in-year cuts to the vital local services that are supporting communities through this crisis and the national effort to beat this deadly disease."
Section 114 reports are issued when a council cannot achieve a balanced budget.
'A problem coming over the hill'
Speaking to the BBC, the Conservative leader of Staffordshire County Council Philip Atkins acknowledged that council finances "will be stretched".
He says the virus has cost his council an estimated £50m - and the local authority says it has so far spent £20m supporting the local care sector.
"The government has given us £38m to help us cover the costs, so there's a bit of a gap," he said.
"There's a problem coming over the hill that we have to be prepared for, but I'm hopeful that the majority of the cost will be paid for by the government."