Vulnerable people and those most affected by the coronavirus outbreak are being offered help to pay their council tax.
Support ranges from deferred payments to discounts for those on low incomes.
A petition on the Parliament website calling for council tax to be scrapped during the duration of the crisis has attracted almost 100,000 signatures.
Blanket deferrals or non-payment would reduce cash for the frontline response to the crisis, larger councils said.
The County Councils Network, which represents 36 of England's county-level local authorities, said it "recognised and understood the motivation behind these schemes...for the most vulnerable and those in the greatest need".
"However, a blanket deferral or non-payment scheme offered across the board could create significant additional financial challenges for local government, reducing cashflow for vital services such as social care which are at the frontline in responding to coronavirus", the network's chairman, Councillor David Williams, said.
Annual bills are usually paid in 10 instalments over 12 months, from April to January.
Under plans being drawn up by some town halls, residents will be able to defer their first payment until July.
Councils offering a deferred payment include Manchester, Oldham, Stockport, Rochdale, Tameside, Trafford, Swindon and Wiltshire.
Not all councils are prepared to offer a deferred payment.
Bristol Mayor Marvin Rees said it was important to maintain Bristol City Council's finances during the crisis.
He said the authority was the only one out of the UK's core cities to retain a 100% council tax reduction scheme.
Most large councils were already due to raise their bills by the full amount allowed before the pandemic struck.
Some councils said they would make use of a £500m hardship fund given by government to local authorities in England to help vulnerable people in their areas.
Leeds and Worcester city councils said those already in receipt of working age council tax support can receive a one-off reduction of up to £150 this year.
Chelmsford City Council said if people were unable to pay council tax due on 8 April, they should contact banks directly to stop the payment.
Councillor Stephen Robinson, its leader, said: "It is not possible to apply a deferral across the board.
"The vast majority of council tax is used to pay police officers and staff, firefighters, care home workers, home care providers, education support workers, and other staff who keep us all safe.
"Applying relief across the board would make it impossible to keep public services going."
Councillor Richard Watts, from the Local Government Association, said: "We are pleased that councils will now be able to provide much-needed support to many households on the lowest incomes by quickly reducing or removing the need for them to pay council tax."
Reporting team: Alex Homer, Daniel Wainwright, the Local Democracy Reporting Service