Ministers are facing competing demands from councils across England for a £1.6bn emergency coronavirus fund.
County councils are seeking assurances that up to £1bn in added social care costs will be met to support nursing homes, recruit carers, source equipment and relieve pressure on the NHS.
But district councils have warned they risk financial collapse if the new money is not allocated "fairly".
The government said councils were getting "significant support".
Earlier this month, Local Government Secretary Robert Jenrick announced a second tranche of emergency funding for councils as part of a £3.2bn package to fund higher service costs caused by the pandemic.
But the Local Government Association, which represents more than 300 councils of different sizes in England and Wales, said local authorities could require four times that in order to make it through.
With the government yet to make clear how the money will be allocated, the BBC's political correspondent Alex Forsyth said many councils were fearful that they could lose out, showing just how stretched services were.
County councils and unitary authorities, which are legally responsible for providing care for the elderly in their areas, say they ought to be prioritised as they are facing an added £1.3bn financial burden,
The County Councils Network, which represents 36 county councils and unitary authorities, has warned already stretched social care budgets could come under further strain if funding mechanisms are "dramatically altered".
They say their members are liable for an estimated extra £322m to meet the escalating costs of care and nursing homes while £259m could be needed to meet growing demand for other services, vital to ensuring people could be treated in the community and not put added strain on NHS beds.
It warned it might have to take "unpalatable decisions" and potentially reduce support for "life-saving" services, including providing testing in care homes, helping to "shield" the most vulnerable in their homes and meeting shortages of personal protective equipment locally.
It also wants the government, separately, to underwrite the cost of potentially billions in lost revenues due to the anticipated steep fall in council tax receipts, income from business rates and other charges.
"It is clear from the evidence provided to ministers that social care is adding the most strain to council budgets as we battle the spread of the disease locally," said Councillor David Williams, the body's chairman.
He added: "If the government changes the distribution of emergency funding it will disproportionately and unfairly impact those councils responsible for adult and children's social care."
Care England, the largest representative body for independent providers of adult social care, has warned of "severe" financial implications from the pandemic and said resources must quickly reach the frontline.
But the 192 district councils in England, which provide services from bin collections to helping the homeless, say they are facing the toughest squeeze on their budgets of any local authorities due to the economic impact of the lockdown and their needs cannot be overlooked.
They warned that income from car parking, leisure facilities, planning and licensing applications is "plummeting" while financial liabilities could dwarf their finances should council tax collection rates fall sharply.
The District Councils Network has called for each district to receive minimum amount of temporary funding, supplemented with additional funding based on the number of residents and businesses in each area.
"We cannot run services on fresh air," said its chair John Fuller. "Districts need to see a significant share of the additional £1.6bn to prevent them winding down services when people and businesses need them most.
"District councils are the ones who are delivering the final mile of public services and have stretched every sinew of resource to step up to the plate and respond to this pandemic."
The Department of Communities and Local Government said it would outline where the £1.6bn would be spent as soon as possible but insisted it was standing by councils at a highly challenging time for them.
"The secretary of state has announced a total of £3.2 billion of funding for councils to support their response to the pandemic," a spokesman said.
"This is a significant package of support which responds to the range of pressures councils have told us they are facing and will support them to protect vital services."