The shortfall comes from a gap of £360 million in this financial year (2020/21) and the need for an estimated £770m in extra funds to deal with the impact of the pandemic next financial year.
The stark picture is set out in a document published by local government body Cosla in a submission to Scottish Government ministers ahead of the coming budget for next year.
The impact of restrictions has meant that vital revenue sources for councils such as parking, libraries, museums, art galleries and theatres, sports facilities, planning and school food has been badly hit and is unlikely to recover until later this year.
Cosla Resources spokeswoman Gail Macgregor said the trend of recent funding settlements for councils had to change.
She said: “This year, across every community in Scotland, local government’s essential role has been magnified and once again we have delivered for our communities.
“Nobody in Scotland has been unaffected by this pandemic and the financial impacts of Covid-19 are severe. Individuals, families and businesses have all felt the effects and continue to look to councils for support every day.
“Sustaining this lifeline support is placing extreme pressure on already strained budgets and without fair funding for local government this year, the consequences for the most vulnerable in our communities would be unacceptable.
“That is why we need fair funding for 2021/22 that respects our communities. Without this, there will be further cuts to services, reductions in spending locally, increases in the inequalities exposed by the pandemic and a much slower recovery.”
Councils say that about about £1bn of additional revenue from the £8.6bn of Treasury cash allocated for Scotland has been “routed” to them by the Scottish Government.
But this will still leave a £360m "gap" this year, according to the document produced on Tuesday. In addition, it says the budget next year will have to rise from a base of £10.7bn this year to £12.1bn next year, to take into account known pressures and Scottish Government commitments, as well as £770m for Covid-related funding.
Cosla president, Councillor Alison Evison, said: “Local government’s role on behalf of our communities cannot be underestimated anymore. The Covid pandemic has shown exactly how much the public rely on us as leaders and as providers of vital services.
“The reality is that in recent budgets, the Scottish Government has chosen not to provide enough funding for the essential services that communities rely on day in day out.
“On top of this, this year we have had to contend with Covid-19 which has seen the inequality in our society grow.
“Our ability to recover from this and continue to deliver for Scotland’s communities depends on a change of emphasis from Scottish Government that provides fair funding for council services.
“If we are to truly recover from this pandemic, then local authorities must receive a fair settlement.”