It said his move was 'clearly a response to the growing opposition of a fundamentally flawed unworkable bill'.
Mr Milburn plans to transfer£100m a year from the NHS to pay for the extra costs councils will face as a result of meeting their responsibilities under the Community Care (Delayed Discharges) Bill (LGC, 22 November).
The Department of Health said the figure was determined after calculations showed it was the maximum councils would have to pay hospitals if they failed to improve services for the long-term ill.
But Alison King (Con), chair of the LGA's social affairs and health executive, said the LGA remained 'fundamentally opposed' to the prospect of councils being fined for delayed discharges set out in the bill.
She said: 'Clearly cash-strapped councils will use the£100m to tackle delayed discharges but, if they are fined, some or all of that money will be transferred back to the DoH. The system could become a costly, bureaucratic nightmare.'
She added: 'It is clear that Mr Milburn is obviously having second thoughts about his bill. If he truly does not want to punish councils, why does he not just simply scrap this ill-thought-out proposal? This£100m does little to help fund the already huge gap in councils' social services budgets.'
The Association of Directors of Social Services welcomed the extra cash.