The Convention of Scottish Local Authorities (Cosla) is in turmoil after three councils last week proposed an overhaul of its constitution, a move seen by Labour as a "Trojan horse" aimed at breaking its grip on the organisation.
Several Labour-led councils, including Glasgow, are understood to be furious at the plan, with some suggesting they might quit Cosla.
A break-up could have profound effects on the funding package for council services, which Cosla negotiates with the Scottish Government, and on wage deals for 250,000 council staff, which are also negotiated through Cosla.
Because it now helps to run 16 of the 32 councils, Labour controls half the vote in the monthly Cosla leaders' meeting, enabling it to block any proposal it dislikes.
But the party fears other councils want to shift power within Cosla to its quarterly convention, where Labour can still be out-voted.
The power struggle ignited in September, when Cosla officials recommended a routine revision of the formula used to distribute Government funding in 2015-16.
As a revised formula would mean some Labour councils suffering cuts, the Labour group voted as a block to reject the proposal, meaning the formula stayed unchanged. The move infuriated non-Labour councils, which wanted a fresh funding deal and more money.
Scottish Borders, which is run by an SNP, Liberal Democrat and Independent coalition, tried to have the decision recalled by last month's convention, but to no avail, as the monthly leaders' meeting is the prime decision-making body.
Scottish Borders, Conservative-led Moray and SNP-led Highland councils have now asked for Cosla's constitution to be reviewed.
The prime mover behind the idea is David Parker, Independent leader of Scottish Borders, and leader of the Independent group in Cosla.
His plan for a working group on the constitution will go to Cosla's convention - where Labour is outnumbered - next month.
Stephen McCabe, secretary of the Labour group within Cosla and leader of Inverclyde Council, said Parker was "playing a dangerous game" and urged him to "pull his tanks back" and enter into a dialogue with the Labour group.
McCabe said: "This is a crisis in the making if [Parker] pursues this line of action. That crisis could potentially be the withdrawal of councils from Cosla."
Parker said talk of splits was "over-exaggerated paranoia" and accused Labour of "playing politics" with the funding distribution formula for councils.
He said: "There's about nine authorities, in effect, telling the rest of Scotland what should happen with distribution - that just can't be right. We need to revisit the constitution because it just doesn't work."
Rory Mair, Cosla chief executive, admitted there was "tension" between the political groups.
He said: "We have to be careful how we handle this tension, so that we retain the cohesion that Cosla has always had. No-one has said to me that Cosla looks like it's about to pull itself apart."