The Convention of Scottish Local Authorities, giving evidence to the parliament's education committee on Wednesday, said reductions in P1-3 class sizes would not be met in Scotland during this parliament - and may not be in the lifetime of the next two parliaments.
All they had agreed to do in the new concordat between local and national government was to make "year-on-year progress" towards the target, Cosla's representatives said.
Karen Whitefield, convener of the education committee, asked Fiona Hyslop, the Education Secretary, if she was satisfied with that. Ms Hyslop admitted that, given the difficulties indicated by Cosla in "delivering every single class", she did not expect the Government to have fulfilled its manifesto promise to cut class sizes in the "lifetime of this parliament".
What she did expect, she said, was that all local authorities "make progress" towards smaller classes.
Later, in a parliamentary debate on class sizes, Ms Hyslop acknowledged that flexibility was needed and it would be easier for some authorities with falling rolls to meet the target than others, such as East and West Lothian, where pupil numbers were rising.
Mary Mulligan, Labour MSP for Linlithgow, said the implication from Cosla's evidence to the education committee was that class sizes would be reduced only where demographics allowed.
She therefore called on Ms Hyslop, a list MSP for the Lothians, to give a guarantee to the people in Linlithgow in West Lothian that they, too, would see class sizes reduced to 18.
Ms Hyslop said the SNP-led West Lothian Council was committed to delivering on class sizes. She expected every other authority to do the same, although she wanted them to focus on the children in the most deprived areas.