The move to feed 165,000 young Scots in their first three years of school has been backed by £64m in public funding, but councils say the true cost could be as much as £88m.
As a result, they warn that trying to deliver the Scottish Government policy on the current timescale - a legal requirement - will inevitably lead to drastic cuts to other frontline services.
Local authority leaders also attacked ministers for failing to consult with councils, with officials saying the plan had been "dumped" on them.
The stark warning comes just weeks after First Minister Alex Salmond announced all pupils in P1 to P3 would get a free lunch from January next year.
Mr Salmond said the move, which matches a plan being introduced in England, would boost health and was worth £330 a year per child to families.
The attack marks a significant new low in relations between Cosla, the umbrella body representing Scotland's councils, and the Scottish Government, with councils feeling increasingly bypassed when major policy decisions are announced.
It also comes at a time when Cosla has been increasingly fragmented due to a rebellion by Labour authorities following a struggle over how decisions are made and where power in the organisation lies.
Cosla president David O'Neill said: "We now have a law that says this policy will be delivered, but we simply don't know how that is going to take place.
"This is supposed to be up and running in January, but we cannot say we can deliver that because of the significant funding shortfall running into millions of pounds.
"We cannot see how this can be delivered, but if councils have to do it then they will have to take money away from other services and that would have a significant impact."
Mr O'Neill said the issues could have been resolved if the policy had been delivered in partnership with the Scottish Government, but that had not taken place. He also suggested there was some £15m in the Scottish Government's budget as a consequence of the free school meals policy being delivered in England.
He added: "We will be engaging with the Scottish Government to discuss how that money can be used to help us deliver this policy."
The row blew up after the Scottish Government estimated the cost of the free school meals policy as £70m with an additional £6m capital investment required in dinner halls and catering services.
Cosla, however, argues the Government has significantly underestimated the probable take-up rate of meals, amounting to a shortfall of some £8m. Cosla also believes it will cost councils £10m in capital costs.
Also, in funding the policy, ministers have assumed that local government will continue to subsidise school meals with £12m of its own funding - which Cosla argues is increasingly untenable in a climate of cuts.
The Scottish Government has come under pressure in recent months to press ahead with the policy from children's charities, anti-poverty campaigners, teaching unions and the Church of Scotland.
A Scottish Government study into free school meals pilots published in 2008 concluded: "Universal healthy free school meals have the potential to impact on children's health, well-being, and educational performance in the medium to long term."
Currently, 33,000 P1-P3 pupils in Scotland are eligible for free school meals.
A Scottish Government spokesman said: "The Scottish Government is fully funding its commitment to ensuring that all children in primary 1 to 3 will have the option of a free school meal from January 2015.
"To deliver our commitment on free school meals, we have been working closely with Cosla and will continue to do so over the next year to explore how any practical implications can be addressed, building on the experience of the free school meal trial over 2007-08."