Speaking at the ADCS annual conference in Manchester, in front of an audience that included Education Secretary Damian Hinds and children's minister Nadhim Zahawi, Gallimore criticised the "intellectually dubious" notion that variations in children's services spending between different councils indicates room for further efficiencies, a case that has previously been put forward by the Department for Education.
"I know there are doubters who think that there's enough money in the system and the problem is it's just spent disproportionately on care placements instead of earlier help and support," he said.
"I don't agree. Nor do I agree with the intellectually dubious notion that if there are variations in costs that there must be room for efficiencies.
"Conference, be very clear - there is not enough money in the system, full stop. There is simply no fat left to trim, instead authorities up and down the country have found themselves having to cut back on early help services which makes no financial sense."
Gallimore went on to describe the notion that the future of local government funding might be fairer if more of it was generated through the retention of business rates or new homes bonus as "frankly bonkers", as it would increase the disparity in local and regional funding.
"Spending money on children and young people, investing in the skills they need for later life, can combat the effects of disadvantage and improve social mobility," Gallimore said.
The LGA has previously warned that annual spending by councils on children's services will rise by £2bn by 2020.
Gallimore called for some of the money the NHS spends on children's community healthcare to be handed to local authorities to ensure it is better spent. He pointed to the estimated £9bn of the annual NHS budget of £100bn being spent on children - the majority on hospital services - as a sign of an imbalance on children's health spending that needs to be redressed.
"[Nine per cent of the NHS budget going on children's health] doesn't seem like the right apportionment to me, because addressing children's health needs early, particularly their mental and emotional health, equals early help to break the generational cycles of adult disadvantage," Gallimore said.
"I'd like to strongly encourage the Department for Education to open discussions with the Department of Health & Social Care about transferring some of the NHS's budget for children's community healthcare to local authorities. I reckon we could do a damn good job of it - and government would be better sighted on how the money was being spent."
Gallimore also warned of the impact of the effective removal of local government from education management.
"The publicly accountable local authority with its historic and democratic legitimacy and its effective record as a school improvement partner has been sidelined - this is not good for children or their families," he said.